research article a general result on the mean integrated...
Post on 14-Jul-2020
0 Views
Preview:
TRANSCRIPT
Research ArticleA General Result on the Mean Integrated SquaredError of the Hard Thresholding Wavelet Estimator under120572-Mixing Dependence
Christophe Chesneau
Laboratoire de Matheematiques Nicolas Oresme Universite de Caen BP 5186 14032 Caen Cedex France
Correspondence should be addressed to Christophe Chesneau christophechesneaugmailcom
Received 10 April 2013 Accepted 22 October 2013 Published 19 January 2014
Academic Editor Zhidong Bai
Copyright copy 2014 Christophe Chesneau This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited
We consider the estimation of an unknown function119891 for weakly dependent data (120572-mixing) in a general setting Our contributionis theoretical we prove that a hard thresholding wavelet estimator attains a sharp rate of convergence under the mean integratedsquared error (MISE) over Besov balls without imposing too restrictive assumptions on the model Applications are given for twotypes of inverse problems the deconvolution density estimation and the density estimation in a GARCH-typemodel both improveexisting results in this dependent context Another application concerns the regression model with random design
1 Introduction
A general nonparametric problem is adopted we aim toestimate an unknown function 119891 via 119899 random variables119881
1 119881
119899from a strictly stationary stochastic process (119881
119905)119905isinZ
We suppose that (119881119905)119905isinZ has a weak dependence structure
the 120572-mixing case is considered This kind of dependencenaturally appears in numerous models as Markov chainsGARCH-type models and discretely observed diffusions(see eg [1ndash3]) The problems where 119891 is the density of 119881
1
or a regression function have received a lot of attention Apartial list of related works includes Robinson [4] Roussas[5 6] Truong and Stone [7] Tran [8] Masry [9 10] Masryand Fan [11] Bosq [12] and Liebscher [13]
For an efficient estimation of 119891 many methods canbe considered The most popular of them are based onkernels splines and wavelets In this note we deal withwavelet methods that have been introduced in iidsettingby Donoho and Johnstone [14 15] and Donoho et al [1617] These methods enjoy remarkable local adaptivity againstdiscontinuities and spatially varying degree of oscillationsComplete reviews and discussions on wavelets in statisticscan be found in for example Antoniadis [18] and Hardleet al [19] In the context of 120572-mixing dependence various
wavelet methods have been elaborated for a wide variety ofnonparametric problems Recent developments can be foundin for example Leblanc [20] Tribouley and Viennet [21]Masry [22] Patil and Truong [23] Doosti et al [24] Doostiand Niroumand [25] Doosti et al [26] Cai and Liang [27]Niu and Liang [28] Benatia and Yahia [29] Chesneau [30ndash32] Chaubey and Shirazi [33] and Abbaszadeh and Emadi[34]
In the general dependent setting described above weprovide a theoretical contribution to the performance ofa wavelet estimator based on a hard thresholding Thisnonlinear wavelet procedure has the features to be fullyadaptive and efficient over a large class of functions 119891 (seeeg [14ndash17 35]) Following the spirit of Kerkyacharian andPicard [36] we determine necessary assumptions on (119881
119905)119905isinZ
and the wavelet basis to ensure that the considered estimatorattains a fast rate of convergence under the MISE over Besovballs The obtained rate of convergence often corresponds tothe near optimal one in the minimax sense for the standardiid case The originality of our result is to be general andsharp it can be applied for nonparametricmodels of differentnatures and improves some existing results This fact isillustrated by the consideration of the density deconvolutionestimation problem and the density estimation problem in a
Hindawi Publishing CorporationJournal of Probability and StatisticsVolume 2014 Article ID 403764 12 pageshttpdxdoiorg1011552014403764
2 Journal of Probability and Statistics
GARCH-type model improving ([30] Proposition 51) and([31] Theorem 2) respectively A last part is devoted to theregression model with random design The obtained resultcompletes the one of Patil and Truong [23]
The organization of this note is as follows In the nextsection we describe the considered wavelet setting Thehard thresholding estimator and its rate of convergenceunder the MISE over Besov balls are presented in Section 3Applications of our general result are given in Section 4 Theproofs are carried out in Section 5
2 Wavelets and Besov Balls
In this section we introduce some notations corresponding towavelets and Besov balls
21 Wavelet Basis We consider the wavelet basis on [0 1]constructs from the Daubechies wavelets db2N with 119873 ge 1(see eg [37]) A brief description of this basis is given belowLet 120601 and 120595 be the initial wavelet functions of the familydb2N These functions have the particularity to be compactlysupported and to belong to the class C119886 for 119873 gt 5119886 For any119895 ge 0 we set Λ
119895= 0 2119895 minus 1 and for 119896 isin Λ
119895
120601119895119896
(119909) = 21198952120601 (2119895119909 minus 119896) 120595119895119896
(119909) = 21198952120595 (2119895119909 minus 119896)
(1)
With appropriated treatments at the boundaries thereexists an integer 120591 such that for any integer ℓ ge 120591 B =120601
ℓ119896 119896 isin Λ
ℓ 120595
119895119896 119895 isin N minus 0 ℓ minus 1 119896 isin Λ
119895 is an
orthonormal basis of L2([0 1]) where
L2([0 1])
= 119891 [0 1] 997888rarr R1003817100381710038171003817119891
10038171003817100381710038172 = (int1
0
1003816100381610038161003816119891 (119909)10038161003816100381610038162
119889119909)
12
lt infin
(2)
For any integer ℓ ge 120591 and 119891 isin L2([0 1]) we have thefollowing wavelet expansion
119891 (119909) = sum119896isinΛ ℓ
119888ℓ119896
120601ℓ119896
(119909) +infin
sum119895=ℓ
sum119896isinΛ 119895
119889119895119896
120595119895119896
(119909)
119909 isin [0 1]
(3)
where 119888119895119896
and 119889119895119896
denote the wavelet coefficients of119891 definedby
119888119895119896
= int1
0
119891 (119909) 120601119895119896
(119909) 119889119909 119889119895119896
= int1
0
119891 (119909) 120595119895119896
(119909) 119889119909
(4)
Technical details can be found in for example Cohen et al[38] and Mallat [39]
In the main result of this paper we will investigate theMISE rate of the proposed estimator by assuming that theunknown function of interest 119891 belongs to a wide class offunctions the Besov class Its definition in terms of waveletcoefficients is presented in the following
22 Besov Balls We say that 119891 isin 119861119904
119901119903(119872)with 119904 gt 0 119901 119903 ge 1
and 119872 gt 0 if and only if there exists a constant 119862 gt 0 suchthat the wavelet coefficients of 119891 given by (4) satisfy
2120591(12minus1119901)( sum119896isinΛ 120591
10038161003816100381610038161198881205911198961003816100381610038161003816119901
)
1119901
+ (infin
sum119895=120591
(2119895(119904+12minus1119901)( sum119896isinΛ 119895
10038161003816100381610038161003816119889119895119896
10038161003816100381610038161003816119901
)
1119901
)
119903
)
1119903
le 119862
(5)
with the usualmodifications if119901 = infin or 119903 = infin Note that forparticular choices of 119904 119901 and 119903119861119904
119901119903(119872) contains the classical
Holder and Sobolev balls (see eg [40] and [19])
Remark 1 We have chosen a wavelet basis on [0 1] to fix thenotations wavelet basis on another interval can be consideredin the rest of the study without affecting the results
3 Statistical Framework Estimator and Result
31 Statistical Framework As mentioned in Section 1 anonparametric estimation setting as general as possible isadopted we aim to estimate an unknown function 119891 isin
L2([0 1]) via 119899 random variables (or vectors) 1198811 119881
119899from
a strictly stationary stochastic process (119881119905)119905isinZ defined on a
probability space (ΩAP) We suppose that (119881119905)119905isinZ has a
120572-mixing dependence structure with exponential decay ratethat is there exist two constants 120574 gt 0 and 120579 gt 0 such that
sup119898ge1
(119890120579119898120572119898) le 120574 (6)
where 120572119898
= sup(119860119861)isinF119881
minusinfin0timesF119881119898infin
|P(119860 cap 119861) minus P(119860)P(119861)|F119881
minusinfin0is the 120590-algebra generated by the random variables (or
vectors) 119881minus1
1198810and F119881
119898infinis the 120590-algebra generated by
the random variables (or vectors) 119881119898 119881
119898+1
The 120572-mixing dependence is reasonably weak it is sat-isfied by a wide variety of models including Markov chainsGARCH-type models and discretely observed diffusions(see for instance [1ndash3 41])
The considered estimator for 119891 is presented below
32 Estimator We define the hard thresholding waveletestimator 119891 by
119891 (119909) = sum119896isinΛ 1198950
1198881198950 119896
1206011198950 119896
(119909) +
1198951
sum119895=1198950
sum119896isinΛ 119895
1198891198951198961|119889119895119896|ge120581120582119895
120595119895119896
(119909)
(7)
where
119888119895119896
=1
119899
119899
sum119894=1
119902 (120601119895119896
119881119894) 119889
119895119896=
1
119899
119899
sum119894=1
119902 (120595119895119896
119881119894) (8)
Journal of Probability and Statistics 3
1 is the indicator function 120581 gt 0 is a large enough constant1198950is the integer satisfying
21198950 = [120591 ln 119899] (9)
where [119886] denotes the integer part of 119886 and 1198951is the integer
satisfying
21198951 = [(119899
(ln 119899)3)
1(2120588+1)
] (10)
120582119895= 2120588119895radic ln 119899
119899 (11)
Here it is supposed that there exists a function 119902 L2([0 1])times119881
1(Ω) rarr C such that
(H1) for 120574 isin 120601 120595 any integer 119895 ge 1198950and 119896 isin Λ
119895
E (119902 (120574119895119896
1198811)) = int
1
0
119891 (119909) 120574119895119896
(119909) 119889119909 (12)
where E denotes the expectation(H2) there exist two constants 119862 gt 0 and 120588 ge 0
satisfying for 120574 isin 120601 120595 for any integer 119895 ge 1198950
and 119896 isin Λ119895 (i) sup
119909isin1198811(Ω)|119902(120574
119895119896 119909)| le 119862212058811989521198952 (ii)
E(|119902(120574119895119896
1198811)|2) le 11986222120588119895 (iii) for any 119898 isin 1 119899 minus
1 ge 110038161003816100381610038161003816C119900V (119902 (120574
119895119896 119881
119898+1) 119902 (120574
119895119896 119881
1))
10038161003816100381610038161003816 le 119862221205881198952minus119895 (13)
whereC119900V denotes the covariance that isC119900V(119883 119884) =
E(119883119884)minusE(119883)E(119884) 119884 denotes the complex conjugateof 119884
For well-known nonparametric models in the iid set-ting hard thresholding wavelet estimators and importantresults can be found in for example Donoho and Johnstone[14 15] Donoho et al [16 17] Delyon and Juditsky [35]Kerkyacharian and Picard [36] and Fan and Koo [42] In the120572-mixing context 119891 defined by (7) is a general and improvedversion of the estimator considered in Chesneau [30 31] Themain differences are the presence of the tuning parameter 120588and the global definition of the function 119902 offering numerouspossibilities of applications Three of them are explored inSection 4
Comments on the Assumptions The assumption (H1) ensuresthat (8) are unbiased estimators for 119888
119895119896and 119889
119895119896given by
(4) whereas (H2) is related to their good performance SeeProposition 10These assumptions are not too restrictive Forinstance if we consider the standard density estimation prob-lem where (119881
119905)119905isinZ are iid random variables with bounded
density 119891 the function 119902(120574 119909) = 120574(119909) satisfies (H1) and (H2)with 120588 = 0 (note that thanks to the independence of (119881
119905)119905isinZ
the covariance term in (H2)-(iii) is zero)The technical detailsare given in Donoho et al [17]
Lemma 2 describes a simple situation in which assump-tion (H2)-(iii) is satisfied
Lemma 2 We make the following assumptions
(F1) Let119906 be the density of 1198811and let119906
(1198811 119881119898+1)be the density
of (1198811 119881
119898+1) for any 119898 isin Z We suppose that there
exists a constant 119862 gt 0 such that
sup119898isin1119899minus1Z
sup(119909119910)isin1198811(Ω)times119881119898+1(Ω)
10038161003816100381610038161003816119906(1198811 119881119898+1)(119909 119910)
minus119906 (119909) 119906 (119910)1003816100381610038161003816 le 119862
(14)
(F2) There exist two constants 119862 gt 0 and 120588 ge 0 satisfyingfor 120574 isin 120601 120595 for any integer 119895 ge 119895
0and 119896 isin Λ
119895
int1198811(Ω)
10038161003816100381610038161003816119902 (120574119895119896
119909)10038161003816100381610038161003816 119889119909 le 11986221205881198952minus1198952 (15)
Then under (F1) and (F2) (H2)-(iii) is satisfied
33 Result Theorem 3 determines the rate of convergenceattained by 119891 under the MISE over Besov balls
Theorem 3 We consider the general statistical settingdescribed in Section 31 Let 119891 be (7) under (H1) and (H2)Suppose that 119891 isin 119861119904
119901119903(119872) with 119903 ge 1 119901 ge 2 and 119904 isin (0119873)
or 119901 isin [1 2) and 119904 isin ((2120588 + 1)119901119873) Then there exists aconstant 119862 gt 0 such that
E (10038171003817100381710038171003817119891 minus 119891
100381710038171003817100381710038172
2) le 119862(
ln 119899
119899)
2119904(2119904+2120588+1)
(16)
The rate of convergence ldquo((ln 119899)119899)2119904(2119904+2120588+1)rdquo is oftenthe near optimal one in the minimax sense for numerousstatistical problems in a iid setting (see eg [19 43])Moreover note that Theorem 3 is flexible the assumptionson (119881
119905)119905isinZ related to the definition of 119902 in (H1) and (H2)
are mild In the next section this flexibility is illustrated forthree sophisticated nonparametric estimation problems thedensity deconvolution estimation problem the density esti-mation problem in a GARCH-typemodel and the regressionfunction estimation in the regression model with randomdesign
4 Applications
41 Density Deconvolution Let (119881119905)119905isinZ be a strictly stationary
stochastic process such that
119881119905= 119883
119905+ 120598
119905 119905 isin Z (17)
where (119883119905)119905isinZ is a strictly stationary stochastic process with
unknown density119891 and (120598119905)119905isinZ is a strictly stationary stochas-
tic process with known density 119892 It is supposed that 120598119905and
119883119905are independent for any 119905 isin Z and (119881
119905)119905isinZ is a 120572-mixing
process with exponential decay rate (see Section 31 for aprecise definition) Our aim is to estimate 119891 via 119881
1 119881
119899
from (119881119905)119905isinZ Some related works are Masry [44] Kulik [45]
Comte et al [46] and Van Zanten and Zareba [47]We formulate the following assumptions
4 Journal of Probability and Statistics
(G1) The support of 119891 is [0 1](G2) There exists a constant 119862 gt 0 such that
sup119909isinR
119891 (119909) le 119862 lt infin (18)
(G3) Let 119906 be the density of1198811We suppose that there exists
a constant 119862 gt 0 such that
sup119909isinR
119906 (119909) le 119862 (19)
(G4) For any 119898 isin Z let 119906(1198811 119881119898+1)
be the density of(119881
1 119881
119898+1) We suppose that there exists a constant
119862 gt 0 such that
sup119898isinZ
sup(119909119910)isinR2
119906(1198811 119881119898+1)
(119909 119910) le 119862 (20)
(G5) For any integrable function 120574 we define its Fouriertransform by
F (120574) (119909) = intinfin
minusinfin
120574 (119910) 119890minus119894119909119910119889119910 119909 isin R (21)
We suppose that there exist three known constants119862 gt 0119888 gt 0 and 120575 gt 1 such that for any 119909 isin R
(i) the Fourier transform of 119892 satisfies
1003816100381610038161003816F (119892) (119909)1003816100381610038161003816 ge
119888
(1 + 1199092)1205752
(22)
(ii) for any ℓ isin 0 1 2 the ℓth derivative of the Fouriertransform of 119892 satisfies
100381610038161003816100381610038161003816(F (119892) (119909))
(ℓ)100381610038161003816100381610038161003816le
119862
(1 + |119909|)120575+ℓ (23)
We are now in the position to present the result
Theorem 4 We consider the model (17) Suppose that (G1)ndash(G5) are satisfied Let 119891 be defined as in (7) with
119902 (120574 119909) =1
2120587int
infin
minusinfin
F (120574) (119910)
F (119892) (119910)119890minus119894119910119909119889119910 (24)
whereF(120574)(119910) denotes the complex conjugate ofF(120574)(119910) and120588 = 120575 (appearing in(G5))
Suppose that 119891 isin 119861119904
119901119903(119872) with 119903 ge 1 119901 ge 2 and 119904 isin
(0119873) or 119901 isin [1 2) and 119904 isin ((2120575 + 1)119901119873) Then thereexists a constant 119862 gt 0 such that
E (10038171003817100381710038171003817119891 minus 119891
100381710038171003817100381710038172
2) le 119862(
ln 119899
119899)
2119904(2119904+2120575+1)
(25)
Theorem 4 improves ([30] Proposition 51) in terms ofrate of convergence we gain a logarithmic term
Moreover it is established that in the iid settingldquo((ln 119899)119899)2119904(2119904+2120575+1)rdquo is
(i) exactly the rate of convergence attained by the hardthresholding wavelet estimator
(ii) the near optimal rate of convergence in the minimaxsense
The details can be found in Fan and Koo [42] ThusTheorem 4 can be viewed as an extension of this existingresult to the weak dependent case
42 GARCH-TypeModel We consider the strictly stationarystochastic process (119881
119905)119905isinZ where for any 119905 isin Z
119881119905= 1205902
119905119885
119905 (26)
(1205902
119905)119905isinZ is a strictly stationary stochastic process with
unknown density 119891 and (119885119905)119905isinZ is a strictly stationary
stochastic process with known density 119892 It is supposed that1205902
119905and 119885
119905are independent for any 119905 isin Z and (119881
119905)119905isinZ is a 120572-
mixing process with exponential decay rate (see Section 31for a precise definition) Our aim is to estimate 119891 via119881
1 119881
119899from (119881
119905)119905isinZ Some related works are Comte et al
[46] and Chesneau [31]We formulate the following assumptions(J1) There exists a positive integer 120575 such that
119892 (119909) =1
(120575 minus 1)(minus ln119909)
120575minus1 119909 isin [0 1] (27)
Let us remark that 119892 is the density of prod120575
119894=1119880
119894 where
1198801 119880
120575are 120575 iid random variables having the
common distributionU([0 1])(J2) The support of 119891 is [0 1] and 119891 isin L2([0 1])(J3) Let 119906 be the density of119881
1We suppose that there exists
a constant 119862 gt 0 such thatsup119909isinR
119906 (119909) le 119862 (28)
(J4) For any 119898 isin Z let 119906(1198811 119881119898+1)
be the density of(119881
1 119881
119898+1) We suppose that there exists a constant
119862 gt 0 such thatsup119898isinZ
sup(119909119910)isinR2
119906(1198811 119881119898+1)
(119909 119910) le 119862 (29)
We are now in the position to present the result
Theorem 5 We consider model (26) Suppose that (J1)ndash(J4)are satisfied Let 119891 be defined as in (7) with
119902 (120574 119909) = 119879120575(120574) (119909) (30)
where for any positive integer ℓ 119879(120574)(119909) = (119909120574(119909))1015840 and119879
ℓ(120574)(119909) = 119879(119879
ℓminus1(120574))(119909) and 120588 = 120575 (appearing in (J1))
Suppose that 119891 isin 119861119904
119901119903(119872) with 119903 ge 1 119901 ge 2 and 119904 isin
(0119873) or 119901 isin [1 2) and 119904 isin ((2120575 + 1)119901119873) Then thereexists a constant 119862 gt 0 such that
E (10038171003817100381710038171003817119891 minus 119891
100381710038171003817100381710038172
2) le 119862(
ln 119899
119899)
2119904(2119904+2120575+1)
(31)
Theorem 5 significantly improves ([31] Theorem 2) interms of rate of convergence we gain an exponent 12
Journal of Probability and Statistics 5
43 Nonparametric Regression Model We consider thestrictly stationary stochastic process (119881
119905)119905isinZ where for any
119905 isin Z 119881119905= (119884
119905 119883
119905)
119884119905= 119891 (119883
119905) + 120585
119905 (32)
(119883119905)119905isinZ is a strictly stationary stochastic process with
unknown density 119892 (120585119905)119905isinZ is a strictly stationary centered
stochastic process and119891 is the unknown regression functionIt is supposed that 119883
119905and 120585
119905are independent for any 119905 isin Z
and (119881119905)119905isinZ is a 120572-mixing process with exponential decay
rate (see Section 31 for a precise definition) Our aim is toestimate 119891 via 119881
1 119881
119899from (119881
119905)119905isinZ Applications of this
problem can be found in Hardle [48] Wavelet methods canbe found in Patil and Truong [23] Doosti et al [24] Doostiet al [26] and Doosti and Niroumand [25]
We formulate the following assumptions
(K1) The support of 119891 and 119892 is [0 1] and 119891 and 119892 isin
L2([0 1])(K2) 120585
1(Ω) is bounded
(K3) There exists a constant 119862 gt 0 such that
sup119909isin[01]
1003816100381610038161003816119891 (119909)1003816100381610038161003816 le 119862 (33)
(K4) There exist two constants 119888lowastgt 0 and 119862 gt 0 such that
119888lowastle inf
119909isin[01]
119892 (119909) sup119909isin[01]
119892 (119909) le 119862 (34)
(K5) Let 119906 be the density of1198811We suppose that there exists
a constant 119862 gt 0 such that
sup119909isinRtimes[01]
119906 (119909) le 119862 (35)
(K6) For any 119898 isin Z let 119906(1198811 119881119898+1)
be the density of(119881
1 119881
119898+1) We suppose that there exists a constant
119862 gt 0 such that
sup119898isinZ
sup(119909119910)isin(Rtimes[01])times(Rtimes[01])
119906(1198811 119881119898+1)
(119909 119910) le 119862 (36)
We are now in the position to present the result
Theorem 6 We consider the model (32) Suppose that (K1)ndash(K6) are satisfied Let 119891 be the truncated ratio estimatorConsider
119891 (119909) =V (119909)119892 (119909)
1|119892(119909)|ge119888lowast2
(37)
where
(i) V is defined as in (7) with
119902 (120574 (119909 119909lowast)) = 119909120574 (119909
lowast) (38)
and 120588 = 0
(ii) 119892 is defined as in (7) with 119883119905instead of 119881
119905
119902 (120574 119909) = 120574 (119909) (39)
and 120588 = 0(iii) 119888
lowastis the constant defined in (K4)
Suppose that 119891119892 isin 119861119904
119901119903(119872) and 119892 isin 119861119904
119901119903(119872) with 119903 ge 1
119901 ge 2 and 119904 isin (0119873) or 119901 isin [1 2) and 119904 isin (1119901119873) Thenthere exists a constant 119862 gt 0 such that
E (10038171003817100381710038171003817119891 minus 119891
100381710038171003817100381710038172
2) le 119862(
ln 119899
119899)
2119904(2119904+1)
(40)
The estimator (37) is derived by combining the procedureof Patil and Truong [23] with the truncated approach ofVasiliev [49]
Theorem 6 completes Patil and Truong [23] in terms ofrates of convergence under the MISE over Besov balls
Remark 7 The assumption(K2) can be relaxed with anotherstrategy to the one developed in Theorem 6 Some technicalelements are given in Chesneau [32]
Conclusion Considering the weak dependent case on theobservations we prove a general result on the rate of conver-gence attains by a hard wavelet thresholding estimator underthe MISE over Besov balls This result is flexible it can beapplied for a wide class of statistical models Moreover theobtained rate of convergence is sharp it can correspond to thenear optimal one in the minimax sense for the standard iidcase Some recent results on sophisticated statistical problemsare improved Thanks to its flexibility the perspectives ofapplications of our theoretical result in other contexts arenumerous
5 Proofs
In this section 119862 denotes any constant that does not dependon 119895 119896 and 119899 Its value may change from one term to anotherand may depend on 120601 or 120595
51 Key Lemmas Let us present two lemmas which will beused in the proofs
Lemma 8 shows a sharp covariance inequality under the120572-mixing condition
Lemma 8 (see [50]) Let (119882119905)119905isinZ be a strictly stationary 120572-
mixing process withmixing coefficient 120572119898119898 ge 0 and let ℎ and
119896 be two measurable functions Let 119901 gt 0 and 119902 gt 0 satisfying1119901 + 1119902 lt 1 such that E(|ℎ(119882
1)|119901) and E(|119896(119882
1)|119902) exist
Then there exists a constant 119862 gt 0 such that1003816100381610038161003816C119900V (ℎ (119882
1) 119896 (119882
119898+1))1003816100381610038161003816
le 1198621205721minus1119901minus1119902
119898(E (
1003816100381610038161003816ℎ (1198821)1003816100381610038161003816119901
))1119901
(E(1003816100381610038161003816119896 (119882
1)1003816100381610038161003816119902
))1119902
(41)
Lemma 9 below presents a concentration inequality for120572-mixing processes
6 Journal of Probability and Statistics
Lemma9 (see [13]) Let (119882119905)119905isinZ be a strictly stationary process
with the 119898th strongly mixing coefficient 120572119898 119898 ge 0 let 119899 be a
positive integer let ℎ R rarr C be a measurable function andfor any 119905 isin Z 119880
119905= ℎ(119882
119905) We assume that E(119880
1) = 0 and
there exists a constant 119872 gt 0 satisfying |1198801| le 119872 Then for
any 119898 isin 1 [1198992] and 120582 gt 0 we have
P(1
119899
100381610038161003816100381610038161003816100381610038161003816
119899
sum119894=1
119880119894
100381610038161003816100381610038161003816100381610038161003816ge 120582)
le 4 exp(minus1205822119899
16 (119863119898119898 + 1205821198721198983)
) + 32119872
120582119899120572
119898
(42)
where
119863119898
= max119897isin12119898
V (119897
sum119894=1
119880119894) (43)
52 Intermediary Results
Proof of Lemma 2 Using a standard expression of the covari-ance and (F1) as well as(F2) we obtain
10038161003816100381610038161003816C119900V (119902 (120574119895119896
119881119898+1
) 119902 (120574119895119896
1198811))
10038161003816100381610038161003816
=100381610038161003816100381610038161003816100381610038161003816int1198811(Ω)
int1198811(Ω)
119902 (120574119895119896
119909) 119902 (120574119895119896
119910)
times (119906(1198811 119881119898+1)
(119909 119910) minus 119906 (119909) 119906 (119910)) 119889119909119889119910100381610038161003816100381610038161003816100381610038161003816
le int1198811(Ω)
int1198811(Ω)
10038161003816100381610038161003816119902 (120574119895119896
119909)1003816100381610038161003816100381610038161003816100381610038161003816119902 (120574
119895119896 119910)
10038161003816100381610038161003816
times10038161003816100381610038161003816119906(1198811 119881119898+1)
(119909 119910) minus 119906 (119909) 119906 (119910)10038161003816100381610038161003816 119889119909 119889119910
le 119862(int1198811(Ω)
10038161003816100381610038161003816119902 (120574119895119896
119909)10038161003816100381610038161003816 119889119909)
2
le 119862221205881198952minus119895
(44)
This ends the proof of Lemma 2
Proposition 10 proves probability and moments inequal-ities satisfied by the estimators (8)
Proposition 10 Let 119895119896
and 120573119895119896
be defined as in (8) under(H1) and (H2) let 119895
0be (9) and let 119895
1be (10)
(a) There exists a constant 119862 gt 0 such that for any 119895 isin119895
0 119895
1 and 119896 isin Λ
119895
E (10038161003816100381610038161003816119888119895119896 minus 119888
119895119896
100381610038161003816100381610038162
) le 11986222120588119895 1
119899 (45)
E (10038161003816100381610038161003816119889119895119896
minus 119889119895119896
100381610038161003816100381610038162
) le 11986222120588119895 1
119899 (46)
(b) There exists a constant 119862 gt 0 such that for any 119895 isin119895
0 119895
1 and 119896 isin Λ
119895
E (10038161003816100381610038161003816119889119895119896
minus 119889119895119896
100381610038161003816100381610038164
) le 11986224120588119895 (47)
(c) Let 120582119895be defined as in (11) There exists a constant 119862 gt
0 such that for any 120581 large enough 119895 isin 1198950 119895
1 and
119896 isin Λ119895 we have
P(10038161003816100381610038161003816119889119895119896
minus 119889119895119896
10038161003816100381610038161003816 ge120581120582
119895
2) le 119862
1
1198994 (48)
Proof of Proposition 10 (a) Using (H1) and the stationarity of(119881
119905)119905isinZ we obtain
E (10038161003816100381610038161003816119888119895119896 minus 119888
119895119896
100381610038161003816100381610038162
) = V (119888119895119896
)
=1
1198992
119899
sum119894=1
V (119902 (120601119895119896
119881119894)) +
2
1198992
times119899
sumV=2
Vminus1
sumℓ=1
Re (C119900V (119902 (120601
119895119896 119881V) 119902 (120601
119895119896 119881
ℓ)))
=1
1198992
119899
sum119894=1
V (119902 (120601119895k 119881119894
)) +2
1198992
times119899minus1
sum119898=1
(119899 minus 119898)Re (C119900V (119902 (120601
119895119896 119881
119898+1) 119902 (120601
119895119896 119881
1)))
le1
119899(E (
10038161003816100381610038161003816119902 (120601119895119896
1198811)100381610038161003816100381610038162
)
+2119899minus1
sum119898=1
10038161003816100381610038161003816C119900V (119902 (120601119895119896
119881119898+1
) 119902 (120601119895119896
1198811))
10038161003816100381610038161003816)
(49)
By (H2)-(ii) we get
E (10038161003816100381610038161003816119902(120601119895119896
1198811)100381610038161003816100381610038162
) le 11986222120588119895 (50)
For the covariance term note that
119899minus1
sum119898=1
10038161003816100381610038161003816C119900V (119902 (120601119895119896
119881119898+1
) 119902 (120601119895119896
1198811))
10038161003816100381610038161003816 = 119860 + 119861 (51)
where
119860 =[ln 119899120579]minus1
sum119898=1
10038161003816100381610038161003816C119900V (119902 (120601119895119896
119881119898+1
) 119902 (120601119895119896
1198811))
10038161003816100381610038161003816
119861 =119899minus1
sum119898=[(ln 119899)120579]
10038161003816100381610038161003816C119900V (119902 (120601119895119896
119881119898+1
) 119902 (120601119895119896
1198811))
10038161003816100381610038161003816
(52)
It follows from (H2)-(iii) and 2minus119895 le 2minus1198950 lt 2(ln 119899)minus1 that
119860 le 119862221205881198952minus119895 [ln 119899
120579] le 11986222120588119895 (53)
Journal of Probability and Statistics 7
The Davydov inequality described in Lemma 8 with 119901 = 119902 =
4 (H2)-(i)-(ii) and 2119895 le 21198951 le 119899 give
119861 le 119862radicE (10038161003816100381610038161003816119902(120601119895119896
1198811)100381610038161003816100381610038164
)119899minus1
sum119898=[(ln 119899)120579]
radic120572119898
le 119862212058811989521198952radicE (10038161003816100381610038161003816119902(120601119895119896
1198811)100381610038161003816100381610038162
)infin
sum119898=[(ln 119899)120579]
119890minus1205791198982
= 11986222120588119895radic119899119890minus(ln 119899)2 le 11986222120588119895
(54)
Thus119899minus1
sum119898=1
10038161003816100381610038161003816C119900V (119902 (120601119895119896
119881119898+1
) 119902 (120601119895119896
1198811))
10038161003816100381610038161003816 le 11986222120588119895 (55)
Putting (49) (50) and (55) together the first point in (a)is proved The proof of the second point is identical with 120595instead of 120601
(b) Thanks to (H2)-(i) we have |119889119895119896
| le
sup119909isin1198811(Ω)
|119902(120595119895119896
119909)| le 119862212058811989521198952 It follows from thetriangular inequality and |119889
119895119896| le 119891
2le 119862 that
10038161003816100381610038161003816119889119895119896minus 119889
119895119896
10038161003816100381610038161003816 le10038161003816100381610038161003816119889119895119896
10038161003816100381610038161003816 +10038161003816100381610038161003816119889119895119896
10038161003816100381610038161003816 le 119862212058811989521198952 (56)
This inequality and the second result of (a) yield
E (10038161003816100381610038161003816119889119895119896
minus 119889119895119896
100381610038161003816100381610038164
) le 119862221205881198952119895E (
10038161003816100381610038161003816119889119895119896minus 119889
119895119896
100381610038161003816100381610038162
) le 119862241205881198952119895 1
119899
(57)
Using 2119895 le 21198951 le 119899 the proof of (b) is completed(c) We will use the Liebscher inequality described in
Lemma 9 Let us set
119880119894= 119902 (120595
119895119896 119881
119894) minus E (119902 (120595
119895119896 119881
1)) (58)
We have E(1198801) = 0 and by(H2)-(i) and 2119895 le 21198951 le
119899(ln 119899)3
1003816100381610038161003816119880119894
1003816100381610038161003816 le 2 sup119909isin1198811(Ω)
10038161003816100381610038161003816119902 (120595119895119896
119909)10038161003816100381610038161003816 le 119862212058811989521198952 le 1198622120588119895
radic119899
(ln 119899)3
(59)
(so 119872 = 1198622120588119895radic119899(ln 119899)3)Proceeding as for the proofs of the bounds in (a) for any
integer 119897 le 119862 ln 119899 since 2minus119895 le 2minus1198950 le 2(ln 119899)minus1 we show that
V (119897
sum119894=1
119880119894)
= V (119897
sum119894=1
119902 (120595119895119896
119881119894)) le 11986222120588119895 (119897 + 11989722minus119895) le 11986222120588119895119897
(60)
Therefore
119863119898
= max119897isin12119898
V (119897
sum119894=1
119880119894) le 11986222120588119895119898 (61)
Owing to Lemma 9 applied with 1198801 119880
119899 120582 = 120581120582
1198952
119898 = [radic120581 ln 119899] 119872 = 1198622120588119895radic119899(ln 119899)3 and the bound (61) weobtain
P(10038161003816100381610038161003816119889119895119896
minus 119889119895119896
10038161003816100381610038161003816 ge120581120582
119895
2)
le 119862(exp(minus11986212058121205822
119895119899
119863119898119898 + 120581120582
119895119898119872
) +119872
120582119895
119899119890minus120579119898)
le 119862( exp (minus119862((120581222120588119895 ln 119899)
times (22120588119895 + 1205812120588119895radic (ln 119899)
119899
times [radic120581 ln 119899] 2120588119895
radic119899
(ln 119899)3)
minus1
))
+ radic119899(ln 119899)3
(ln 119899) 119899119899119890minus120579[radic120581 ln 119899])
le 119862(119899minus1198621205812(1+120581
32) + 1198992minus120579radic120581)
(62)
Taking 120581 large enough the last term is bounded by1198621198994Thiscompletes the proof of(c)
This completes the proof of Proposition 10
Proof of Theorem 3 Theorem 3 can be proved by combiningarguments of ([36] Theorem 51) and ([51] Theorem 42) Itis close to ([30] Proof of Theorem 2) by taking 120579 rarr infin Theinterested reader can find the details below
We consider the following wavelet decomposition for 119891
119891 (119909) = sum119896isinΛ 1198950
1198881198950 119896
1206011198950 119896
(119909) +infin
sum119895=1198950
sum119896isinΛ 119895
119889119895119896
120595119895119896
(119909) (63)
where 1198881198950 119896
= int1
0119891(119909)120601
1198950 119896(119909)119889119909 and 119889
119895119896= int
1
0119891(119909)120595
119895119896(119909)119889119909
Using the orthonormality of the wavelet basis B theMISE of 119891 can be expressed as
E (10038171003817100381710038171003817119891 minus 119891
100381710038171003817100381710038172
2) = 119875 + 119876 + 119877 (64)
where
119875 = sum119896isinΛ 1198950
E (100381610038161003816100381610038161198881198950 119896 minus 119888
1198950 119896
100381610038161003816100381610038162
)
119876 =
1198951
sum119895=1198950
sum119896isinΛ 119895
E(100381610038161003816100381610038161003816119889
1198951198961|119889119895119896|ge120581120582119895
minus 119889119895119896
100381610038161003816100381610038161003816
2
)
119877 =infin
sum119895=1198951+1
sum119896isinΛ 119895
1198892
119895119896
(65)
8 Journal of Probability and Statistics
Let us now investigate sharp upper bounds for 119875 119877 and119876 successively
Upper Bound for 119875 The point (a) of Proposition 10 and2119904(2119904 + 2120588 + 1) lt 1 yield
119875 le 11986221198950
119899le 119862
ln 119899
119899le 119862(
ln 119899
119899)
2119904(2119904+2120588+1)
(66)
Upper Bound for 119877(i) For 119903 ge 1 and119901 ge 2 we have119891 isin 119861119904
119901119903(119872) sube 119861119904
2infin(119872)
Using 2119904(2119904 + 2120588 + 1) lt 2119904(2120588 + 1) we obtain
119877 le 119862infin
sum119895=1198951+1
2minus2119895119904 le 119862((ln 119899)3
119899)
2119904(2120588+1)
le 119862(ln 119899
119899)
2119904(2119904+2120588+1)
(67)
(ii) For 119903 ge 1 and 119901 isin [1 2) we have 119891 isin 119861119904
119901119903(119872) sube
119861119904+12minus1119901
2infin(119872) The condition 119904 gt (2120588 + 1)119901 implies
that (119904 + 12 minus 1119901)(2120588 + 1) gt 119904(2119904 + 2120588 + 1) Thus
119877 le 119862infin
sum119895=1198951+1
2minus2119895(119904+12minus1119901)
le 119862((ln 119899)3
119899)
2(119904+12minus1119901)(2120588+1)
le 119862(ln 119899
119899)
2119904(2119904+2120588+1)
(68)
Hence for 119903 ge 1 119901 ge 2 and 119904 gt 0 or 119901 isin [1 2) and119904 gt (2120588 + 1)119901 we have
119877 le 119862(ln 119899
119899)
2119904(2119904+2120588+1)
(69)
Upper Bound for 119876 Adopting the notation119863119895119896
= 119889119895119896
minus 119889119895119896
119876 can be written as
119876 =4
sum119894=1
119876119894 (70)
where
1198761=
1198951
sum119895=1198950
sum119896isinΛ 119895
E (10038161003816100381610038161003816119863119895119896
100381610038161003816100381610038162
1|119889119895119896|ge120581120582119895 |119889119895119896|lt1205811205821198952
)
1198762=
1198951
sum119895=1198950
sum119896isinΛ 119895
E (10038161003816100381610038161003816119863119895119896
100381610038161003816100381610038162
1|119889119895119896|ge120581120582119895 |119889119895119896|ge1205811205821198952
)
1198763=
1198951
sum119895=1198950
sum119896isinΛ 119895
E (1198892
1198951198961|119889119895119896|lt120581120582119895 |119889119895119896|ge2120581120582119895
)
1198764=
1198951
sum119895=1198950
sum119896isinΛ 119895
E (1198892
1198951198961|119889119895119896|lt120581120582119895 |119889119895119896|lt2120581120582119895
)
(71)
Upper Bound for 1198761
+ 1198763 Owing to the inequalities
1|119889119895119896|lt120581120582119895 |119889119895119896|ge2120581120582119895
le 1|119895119896|gt1205811205821198952
1|119889119895119896|ge120581120582119895 |119889119895119896|lt1205811205821198952
le
1|119895119896|gt1205811205821198952
and 1|119889119895119896|lt120581120582119895 |119889119895119896|ge2120581120582119895
le 1|119889119895119896|le2|119895119896|
theCauchy-Schwarz inequality and the points(b) and (c) ofProposition 10 we have
1198761+ 119876
3le 119862
1198951
sum119895=1198950
sum119896isinΛ 119895
E (10038161003816100381610038161003816119863119895119896
100381610038161003816100381610038162
1|119895119896|gt1205811205821198952
)
le 119862
1198951
sum119895=1198950
sum119896isinΛ 119895
(E (10038161003816100381610038161003816119863119895119896
100381610038161003816100381610038164
))12
(P(10038161003816100381610038161003816119863119895119896
10038161003816100381610038161003816 gt120581120582
119895
2))
12
le 1198621
1198992
1198951
sum119895=1198950
2119895(1+2120588) le 1198621
119899le 119862(
ln 119899
119899)
2119904(2119904+2120588+1)
(72)
Upper Bound for 1198762 It follows from the point(a) of
Proposition 10 that
1198762le
1198951
sum119895=1198950
sum119896isinΛ 119895
E (10038161003816100381610038161003816119863119895119896
100381610038161003816100381610038162
) 1|119889119895119896|ge1205811205821198952
le 1198621
119899
1198951
sum119895=j0
22120588119895 sum119896isinΛ 119895
1|119889119895119896|gt1205811205821198952
(73)
Let us now introduce the integer 119895lowastdefined by
2119895lowast = [(119899
ln 119899)
1(2119904+2120588+1)
] (74)
Note that 119895lowastisin 119895
0 119895
1 for 119899 large enough
Then 1198762can be bounded as
1198762le 119876
21+ 119876
22 (75)
where
11987621
= 1198621
119899
119895lowast
sum119895=1198950
22120588119895 sum119896isinΛ 119895
1|119889119895119896|gt1205811205821198952
11987622
= 1198621
119899
1198951
sum119895=119895lowast+1
22120588119895 sum119896isinΛ 119895
1|119889119895119896|gt1205811205821198952
(76)
On the one hand we have
11987621
le 119862ln 119899
119899
119895lowast
sum119895=1198950
2119895(1+2120588) le 119862(ln 119899
119899)
2119904(2119904+2120588+1)
(77)
On the other hand we have the following
Journal of Probability and Statistics 9
(i) For 119903 ge 1 and 119901 ge 2 the Markov inequality and 119891 isin119861119904
119901119903(119872) sube 119861119904
2infin(119872) yield
11987622
le 119862ln 119899
119899
1198951
sum119895=119895lowast+1
22120588119895 1
1205822
119895
sum119896isinΛ 119895
1198892
119895119896le 119862
infin
sum119895=119895lowast+1
sum119896isinΛ 119895
1198892
119895119896
le 119862infin
sum119895=119895lowast+1
2minus2119895119904 le 119862(ln 119899
119899)
2119904(2119904+2120588+1)
(78)
(ii) For 119903 ge 1 119901 isin [1 2) and 119904 gt (2120588 + 1)119901 the Markovinequality 119891 isin 119861119904
119901119903(119872) and (2119904 + 2120588 + 1)(2 minus 119901)2 +
(119904 + 12 minus 1119901 + 120588 minus 2120588119901)119901 = 2119904 imply that
11987622
le 119862ln 119899
119899
1198951
sum119895=119895lowast+1
22120588119895 1
120582119901
119895
sum119896isinΛ 119895
10038161003816100381610038161003816119889119895119896
10038161003816100381610038161003816119901
le 119862(ln 119899
119899)
(2minus119901)2 infin
sum119895=119895lowast+1
2119895120588(2minus119901)2minus119895(119904+12minus1119901)119901
le 119862(ln 119899
119899)
(2minus119901)2
2minus119895lowast(119904+12minus1119901+120588minus2120588119901)119901
le 119862(ln 119899
119899)
2119904(2119904+2120588+1)
(79)
Therefore for 119903 ge 1 119901 ge 2 and 119904 gt 0 or 119901 isin [1 2) and119904 gt (2120588 + 1)119901 we have
1198762le C(
ln 119899
119899)
2119904(2119904+2120588+1)
(80)
Upper Bound for 1198764 We have
1198764le
1198951
sum119895=1198950
sum119896isinΛ 119895
1198892
1198951198961|119889119895119896|lt2120581120582119895
(81)
Let 119895lowastbe the integer (74) Then 119876
4can be bound as
1198764le 119876
41+ 119876
42 (82)
where
11987641
=
119895lowast
sum119895=1198950
sum119896isinΛ 119895
1198892
1198951198961|119889119895119896|lt2120581120582119895
11987642
=
1198951
sum119895=119895lowast+1
sum119896isinΛ 119895
1198892
1198951198961|119889119895119896|lt2120581120582119895
(83)
On the one hand we have
11987641
le 119862
119895lowast
sum119895=1198950
21198951205822
119895= 119862
ln 119899
119899
119895lowast
sum119895=1198950
2119895(1+2120588) le 119862(ln 119899
119899)
2119904(2119904+2120588+1)
(84)
On the other hand we have the following
(i) For 119903 ge 1 and 119901 ge 2 since 119891 isin 119861119904
119901119903(119872) sube 119861119904
2infin(119872)
we have
11987642
leinfin
sum119895=119895lowast+1
sum119896isinΛ 119895
1198892
119895119896le 119862
infin
sum119895=119895lowast+1
2minus2119895119904 le 119862(ln 119899
119899)
2119904(2119904+2120588+1)
(85)
(ii) For 119903 ge 1 119901 isin [1 2) and 119904 gt (2120588 + 1)119901 owing to theMarkov inequality 119891 isin 119861119904
119901119903(119872) and (2119904 + 2120588 + 1)(2 minus
119901)2 + (119904 + 12 minus 1119901 + 120588 minus 2120588119901)119901 = 2119904 we get
11987642
le 119862
1198951
sum119895=119895lowast+1
1205822minus119901
119895sum
119896isinΛ 119895
10038161003816100381610038161003816119889119895119896
10038161003816100381610038161003816119901
= 119862(ln 119899
119899)
(2minus119901)2 1198951
sum119895=119895lowast+1
2119895120588(2minus119901) sum119896isinΛ 119895
10038161003816100381610038161003816119889119895119896
10038161003816100381610038161003816119901
le 119862(ln 119899
119899)
(2minus119901)2 infin
sum119895=119895lowast+1
2119895120588(2minus119901)2minus119895(119904+12minus1119901)119901
le 119862(ln 119899
119899)
(2minus119901)2
2minus119895lowast(119904+12minus1119901+120588minus2120588119901)119901
le 119862(ln 119899
119899)
2119904(2119904+2120588+1)
(86)
So for 119903 ge 1 119901 ge 2 and 119904 gt 0 or 119901 isin [1 2) and 119904 gt(2120588 + 1)119901 we have
1198764le 119862(
ln 119899
119899)
2119904(2119904+2120588+1)
(87)
Putting (70) (72) (80) and (87) together for 119903 ge 1 119901 ge 2and 119904 gt 0 or 119901 isin [1 2) and 119904 gt (2120588 + 1)119901 we obtain
119876 le 119862(ln 119899
119899)
2119904(2119904+2120588+1)
(88)
Combining (64) (66) (69) and (88) we complete theproof of Theorem 3
Proof of Theorem 4 The proof of Theorem 4 is a direct appli-cation ofTheorem 3 under (G1)ndash(G5) the function 119902 definedby (24) satisfies (H1) see ([42] equation (2)) and (H2) (i) see([42] Lemma 6) (ii) see ([42] equation (11)) and (iii) see([30] Proof of Proposition 61) with 120588 = 120575
Proof of Theorem 5 The proof ofTheorem 5 is a consequenceofTheorem 3 under (J1)ndash(J4) the function 119902 defined by (30)satisfies (H1) and (H2) (i)-(ii) see ([31] Proposition 1) and(iii) see ([52] equation (26)) with 120588 = 120575
Proof of Theorem 6 Set V(119909) = 119891(119909)119892(119909) Following themethodology of [49] we have
119891 (119909) minus 119891 (119909) = 119878 (119909) minus 119879 (119909) (89)
10 Journal of Probability and Statistics
where
119878 (119909)
=1
119892 (119909)(V (119909) minus V (119909)
+119891 (119909) (119892 (119909) minus 119892 (119909))) 1|119892(119909)|ge119888lowast2
119879 (119909) = 119891 (119909) 1|119892(119909)|lt119888lowast2
(90)
Using (K3) and the indicator function we have
|119878 (119909)| le 119862 (|V (119909) minus V (119909)| + 1003816100381610038161003816119892 (119909) minus 119892 (119909)1003816100381610038161003816) (91)
It follows from |119892(119909)| lt 119888lowast2 cap |119892(119909)| gt 119888
lowast sube |119892(119909) minus
119892(119909)| gt 119888lowast2 (K3) (K4) and the Markov inequality that
|119879 (119909)| le 1198621|119892(119909)minus119892(119909)|gt119888lowast2
le 1198621003816100381610038161003816119892 (119909) minus 119892 (119909)
1003816100381610038161003816 (92)
The triangular inequality yields
10038161003816100381610038161003816119891 (119909) minus 119891 (119909)10038161003816100381610038161003816 le 119862 (|V (119909) minus V (119909)| + 1003816100381610038161003816119892 (119909) minus 119892 (119909)
1003816100381610038161003816) (93)
The elementary inequality (119886 + 119887)2 le 2(1198862 + 1198872) implies that
E (10038171003817100381710038171003817119891 minus 119891
100381710038171003817100381710038172
2) le 119862 (E (V minus V2
2) + E (
1003817100381710038171003817119892 minus 11989210038171003817100381710038172
2)) (94)
We now bound this two MISEs via Theorem 3
Upper Bound for the MISE of V Under (K1)ndash(K6) thefunction 119902 defined by (38) satisfies the following
(H1) With V instead of 119891 since 1205851and 119883
1are independent
with E(1205851) = 0
E (119902 (120574119895119896
1198811))
= E (1198841120574119895119896
(1198831)) = E (119891 (119883
1) 120574
119895119896(119883
1))
= int1
0
119891 (119909) 120574119895119896
(119909) 119892 (119909) 119889119909 = int1
0
V (119909) 120574119895119896
(119909) 119889119909
(95)
(H2) (i)-(ii)-(iii) with 120588 = 0
(i) since 1198841(Ω) is bounded thanks to (K2) and (K3) say
|1198841| le 119872 with 119872 gt 0 we have
sup(119909119909lowast)isin1198811(Ω)
10038161003816100381610038161003816119902 (120574119895119896
(119909 119909lowast))
10038161003816100381610038161003816
= sup(119909119909lowast)isin[minus119872119872]times[01]
10038161003816100381610038161003816119909120574119895119896(119909
lowast)10038161003816100381610038161003816
le 119872 sup119909lowastisin[01]
10038161003816100381610038161003816120574119895119896(119909
lowast)10038161003816100381610038161003816 le 11986221198952
(96)
(ii) using the boundedness of 1198841(Ω) then (K4) we have
E (10038161003816100381610038161003816119902(120574119895119896
1198811)100381610038161003816100381610038162
) = E (1198842
1(120574
119895119896(119883
1))
2
)
le 119862E ((120574119895119896
(1198831))
2
)
= 119862int1
0
(120574119895119896
(119909))2
119892 (119909) 119889119909
le 119862int1
0
(120574119895119896
(119909))2
119889119909 le 119862
(97)
(iii) using the boundedness of 1198841(Ω) and making the
change of variables 119910 = 2119895119909 minus 119896 we obtain
int1198811(Ω)
10038161003816100381610038161003816119902 (120574119895119896
119909)10038161003816100381610038161003816 119889119909
= (int119872
minus119872
|119909| 119889119909)(int1
0
10038161003816100381610038161003816120574119895119896(119909
lowast)10038161003816100381610038161003816 119889119909lowast
)
= 1198722 int1
0
10038161003816100381610038161003816120574119895119896(119909)
10038161003816100381610038161003816 119889119909 le 1198622minus1198952
(98)
We conclude by applying Lemma 2 with 120588 = 0 (K5) and (K6)imply (F1) and the previous inequality implies (F2)
Therefore assuming that V isin 119861119904
119901119903(119872) with 119903 ge 1 119901 ge 2
and 119904 isin (0119873) or 119901 isin [1 2) and 119904 isin (1119901119873) Theorem 3proves the existence of a constant 119862 gt 0 satisfying
E (V minus V22) le 119862(
ln 119899
119899)
2119904(2119904+1)
(99)
Upper Bound for theMISE of 119892 Under (K1)ndash(K6) proceedingas the previous point we show that the function 119902 defined by(39) satisfies (H1) with 119892 instead of 119891 and 119883
119905instead of 119881
119905
and(H2) (i)-(ii)-(iii) with 120588 = 0Therefore assuming that 119892 isin 119861119904
119901119903(119872) with 119903 ge 1 119901 ge 2
and 119904 isin (0119873) or 119901 isin [1 2) and 119904 isin (1119901119873) Theorem 3proves the existance of a constant 119862 gt 0 satisfying
E (1003817100381710038171003817119892 minus 119892
10038171003817100381710038172
2) le 119862(
ln 119899
119899)
2119904(2119904+1)
(100)
Combining (94) (99) and (100) we end the proof ofTheorem 6
Conflict of Interests
The author declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
The author is thankful to the reviewers for their commentswhich have helped in improving the presentation
Journal of Probability and Statistics 11
References
[1] P DoukhanMixing Properties and Examples vol 85 of LectureNotes in Statistics Springer New York NY USA 1994
[2] M Carrasco and X Chen ldquoMixing and moment properties ofvarious GARCH and stochastic volatility modelsrdquo EconometricTheory vol 18 no 1 pp 17ndash39 2002
[3] R C Bradley Introduction to Strong Mixing Conditions Volume1 2 3 Kendrick Press 2007
[4] P M Robinson ldquoNonparametric estimators for time seriesrdquoJournal of Time Series Analysis vol 4 pp 185ndash207 1983
[5] G G Roussas ldquoNonparametric estimation inmixing sequencesof random variablesrdquo Journal of Statistical Planning and Infer-ence vol 18 no 2 pp 135ndash149 1987
[6] G G Roussas ldquoNonparametric regression estimation undermixing conditionsrdquo Stochastic Processes and their Applicationsvol 36 no 1 pp 107ndash116 1990
[7] Y K Truong and C J Stone ldquoNonparametric function estima-tion involving time seriesrdquo The Annals of Statistics vol 20 pp77ndash97 1992
[8] L H Tran ldquoNonparametric function estimation for time seriesby local average estimatorsrdquoThe Annals of Statistics vol 21 pp1040ndash1057 1993
[9] E Masry ldquoMultivariate local polynomial regression for timeseries uniform strong consistency and ratesrdquo Journal of TimeSeries Analysis vol 17 no 6 pp 571ndash599 1996
[10] E Masry ldquoMultivariate regression estimation local polynomialfitting for time seriesrdquo Stochastic Processes and their Applica-tions vol 65 no 1 pp 81ndash101 1996
[11] E Masry and J Fan ldquoLocal polynomial estimation of regressionfunctions for mixing processesrdquo Scandinavian Journal of Statis-tics vol 24 no 2 pp 165ndash179 1997
[12] D Bosq Nonparametric Statistics for Stochastic Processes Esti-mation and Prediction vol 110 of Lecture Notes in StatisticsSpringer New York NY USA 1998
[13] E Liebscher ldquoEstimation of the density and the regressionfunction under mixing conditionsrdquo Statistics and Decisions vol19 no 1 pp 9ndash26 2001
[14] D L Donoho and I M Johnstone ldquoIdeal spatial adaptation bywavelet shrinkagerdquo Biometrika vol 81 no 3 pp 425ndash455 1994
[15] D L Donoho and I M Johnstone ldquoAdapting to unknownsmoothness via wavelet shrinkagerdquo Journal of the AmericanStatistical Association vol 90 pp 1200ndash1224 1995
[16] D Donoho I Johnstone G Kerkyacharian and D PicardldquoWavelet shrinkage asymptopiardquo Journal of the Royal StatisticalSociety B vol 57 pp 301ndash369 1995
[17] D L Donoho I M Johnstone G Kerkyacharian and DPicard ldquoDensity estimation by wavelet thresholdingrdquo Annals ofStatistics vol 24 no 2 pp 508ndash539 1996
[18] A Antoniadis ldquoWavelets in statistics a review (with discus-sion)rdquo Journal of the Italian Statistical Society vol 6 pp 97ndash1441997
[19] W Hardle G Kerkyacharian D Picard and A TsybakovWavelet Approximation and Statistical Applications vol 129 ofLectures Notes in Statistics Springer New York NY USA 1998
[20] F Leblanc ldquoWavelet linear density estimator for a discrete-timestochastic process 119871
119901-lossesrdquo Statistics and Probability Letters
vol 27 no 1 pp 71ndash84 1996[21] K Tribouley andG Viennet ldquo119871
119901adaptive density estimation in
a alpha-mixing frameworkrdquo Annales de lrsquoinstitut Henri PoincareB vol 34 no 2 pp 179ndash208 1998
[22] E Masry ldquoWavelet-based estimation of multivariate regressionfunctions in besov spacesrdquo Journal of Nonparametric Statisticsvol 12 no 2 pp 283ndash308 2000
[23] P N Patil and Y K Truong ldquoAsymptotics for wavelet basedestimates of piecewise smooth regression for stationary timeseriesrdquo Annals of the Institute of Statistical Mathematics vol 53no 1 pp 159ndash178 2001
[24] H Doosti M Afshari and H A Niroumand ldquoWavelets fornonparametric stochastic regression with mixing stochasticprocessrdquo Communications in Statistics vol 37 no 3 pp 373ndash385 2008
[25] H Doosti and H A Niroumand ldquoMultivariate stochasticregression estimation by wavelet methods for stationary timeseriesrdquo Pakistan Journal of Statistics vol 27 no 1 pp 37ndash462009
[26] H Doosti M S Islam Y P Chaubey and P Gora ldquoTwo-dimensional wavelets for nonlinear autoregressive models withan application in dynamical systemrdquo Italian Journal of Pure andApplied Mathematics no 27 pp 39ndash62 2010
[27] J Cai and H Liang ldquoNonlinear wavelet density estimation fortruncated and dependent observationsrdquo International Journal ofWavelets Multiresolution and Information Processing vol 9 no4 pp 587ndash609 2011
[28] S Niu and H Liang ldquoNonlinear wavelet estimation of condi-tional density under left-truncated and 120572-mixing assumptionsrdquoInternational Journal of Wavelets Multiresolution and Informa-tion Processing vol 9 no 6 pp 989ndash1023 2011
[29] F Benatia and D Yahia ldquoNonlinear wavelet regression functionestimator for censored dependent datardquo Journal Afrika Statis-tika vol 7 no 1 pp 391ndash411 2012
[30] C Chesneau ldquoOn the adaptive wavelet deconvolution of adensity for strong mixing sequencesrdquo Journal of the KoreanStatistical Society vol 41 no 4 pp 423ndash436 2012
[31] C Chesneau ldquoWavelet estimation of a density in a GARCH-type modelrdquo Communications in Statistics vol 42 no 1 pp 98ndash117 2013
[32] C Chesneau ldquoOn the adaptive wavelet estimation of a mul-tidimensional regression function under alpha-mixing depen-dence beyond the standard assumptions on the noiserdquo Com-mentationes Mathematicae Universitatis Carolinae vol 4 pp527ndash556 2013
[33] Y P Chaubey and E Shirazi ldquoOn MISE of a nonlinear waveletestimator of the regression function based on biased data understrong mixing rdquo Communications in Statistics In press
[34] M Abbaszadeh andM Emadi ldquoWavelet density estimation andstatistical evidences role for a garch model in the weighteddistributionrdquo Applied Mathematics vol 4 no 2 pp 10ndash4162013
[35] B Delyon and A Juditsky ldquoOn minimax wavelet estimatorsrdquoApplied and Computational Harmonic Analysis vol 3 no 3 pp215ndash228 1996
[36] G Kerkyacharian and D Picard ldquoThresholding algorithmsmaxisets and well-concentrated basesrdquo Test vol 9 no 2 pp283ndash344 2000
[37] I Daubechies Ten Lectures on Wavelets SIAM 1992[38] A Cohen I Daubechies and P Vial ldquoWavelets on the interval
and fast wavelet transformsrdquo Applied and Computational Har-monic Analysis vol 1 no 1 pp 54ndash81 1993
[39] S Mallat A Wavelet Tour of Signal Processing The sparse waywith contributions fromGabriel Peyralphae ElsevierAcademicPress Amsterdam The Netherlands 3rd edition 2009
12 Journal of Probability and Statistics
[40] Y Meyer Ondelettes et Opalphaerateurs Hermann ParisFrance 1990
[41] V Genon-Catalot T Jeantheau and C Laredo ldquoStochasticvolatility models as hidden Markov models and statisticalapplicationsrdquo Bernoulli vol 6 no 6 pp 1051ndash1079 2000
[42] J Fan and J Koo ldquoWavelet deconvolutionrdquo IEEE Transactionson Information Theory vol 48 no 3 pp 734ndash747 2002
[43] A B Tsybakov Introduction alphaa lrsquoestimation non-paramalphaetrique Springer New York NY USA 2004
[44] E Masry ldquoStrong consistency and rates for deconvolutionof multivariate densities of stationary processesrdquo StochasticProcesses and their Applications vol 47 no 1 pp 53ndash74 1993
[45] R Kulik ldquoNonparametric deconvolution problem for depen-dent sequencesrdquo Electronic Journal of Statistics vol 2 pp 722ndash740 2008
[46] F Comte J Dedecker and M L Taupin ldquoAdaptive densitydeconvolution with dependent inputsrdquo Mathematical Methodsof Statistics vol 17 no 2 pp 87ndash112 2008
[47] H van Zanten and P Zareba ldquoA note on wavelet densitydeconvolution for weakly dependent datardquo Statistical Inferencefor Stochastic Processes vol 11 no 2 pp 207ndash219 2008
[48] W Hardle Applied Nonparametric Regression Cambridge Uni-versity Press Cambridge UK 1990
[49] V A Vasiliev ldquoOne investigation method of a ratios typeestimatorsrdquo in Proceedings of the 16th IFAC Symposium onSystem Identialphacation pp 1ndash6 Brussels Belgium July 2012(in progress)
[50] Y Davydov ldquoThe invariance principle for stationary processesrdquoTheory of Probability and Its Applications vol 15 no 3 pp 498ndash509 1970
[51] C Chesneau ldquoWavelet estimation via block thresholding aminimax study under 119871119901-riskrdquo Statistica Sinica vol 18 no 3pp 1007ndash1024 2008
[52] C Chesneau and H Doosti ldquoWavelet linear density estimationfor a GARCH model under various dependence structuresrdquoJournal of the Iranian Statistical Society vol 11 no 1 pp 1ndash212012
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
2 Journal of Probability and Statistics
GARCH-type model improving ([30] Proposition 51) and([31] Theorem 2) respectively A last part is devoted to theregression model with random design The obtained resultcompletes the one of Patil and Truong [23]
The organization of this note is as follows In the nextsection we describe the considered wavelet setting Thehard thresholding estimator and its rate of convergenceunder the MISE over Besov balls are presented in Section 3Applications of our general result are given in Section 4 Theproofs are carried out in Section 5
2 Wavelets and Besov Balls
In this section we introduce some notations corresponding towavelets and Besov balls
21 Wavelet Basis We consider the wavelet basis on [0 1]constructs from the Daubechies wavelets db2N with 119873 ge 1(see eg [37]) A brief description of this basis is given belowLet 120601 and 120595 be the initial wavelet functions of the familydb2N These functions have the particularity to be compactlysupported and to belong to the class C119886 for 119873 gt 5119886 For any119895 ge 0 we set Λ
119895= 0 2119895 minus 1 and for 119896 isin Λ
119895
120601119895119896
(119909) = 21198952120601 (2119895119909 minus 119896) 120595119895119896
(119909) = 21198952120595 (2119895119909 minus 119896)
(1)
With appropriated treatments at the boundaries thereexists an integer 120591 such that for any integer ℓ ge 120591 B =120601
ℓ119896 119896 isin Λ
ℓ 120595
119895119896 119895 isin N minus 0 ℓ minus 1 119896 isin Λ
119895 is an
orthonormal basis of L2([0 1]) where
L2([0 1])
= 119891 [0 1] 997888rarr R1003817100381710038171003817119891
10038171003817100381710038172 = (int1
0
1003816100381610038161003816119891 (119909)10038161003816100381610038162
119889119909)
12
lt infin
(2)
For any integer ℓ ge 120591 and 119891 isin L2([0 1]) we have thefollowing wavelet expansion
119891 (119909) = sum119896isinΛ ℓ
119888ℓ119896
120601ℓ119896
(119909) +infin
sum119895=ℓ
sum119896isinΛ 119895
119889119895119896
120595119895119896
(119909)
119909 isin [0 1]
(3)
where 119888119895119896
and 119889119895119896
denote the wavelet coefficients of119891 definedby
119888119895119896
= int1
0
119891 (119909) 120601119895119896
(119909) 119889119909 119889119895119896
= int1
0
119891 (119909) 120595119895119896
(119909) 119889119909
(4)
Technical details can be found in for example Cohen et al[38] and Mallat [39]
In the main result of this paper we will investigate theMISE rate of the proposed estimator by assuming that theunknown function of interest 119891 belongs to a wide class offunctions the Besov class Its definition in terms of waveletcoefficients is presented in the following
22 Besov Balls We say that 119891 isin 119861119904
119901119903(119872)with 119904 gt 0 119901 119903 ge 1
and 119872 gt 0 if and only if there exists a constant 119862 gt 0 suchthat the wavelet coefficients of 119891 given by (4) satisfy
2120591(12minus1119901)( sum119896isinΛ 120591
10038161003816100381610038161198881205911198961003816100381610038161003816119901
)
1119901
+ (infin
sum119895=120591
(2119895(119904+12minus1119901)( sum119896isinΛ 119895
10038161003816100381610038161003816119889119895119896
10038161003816100381610038161003816119901
)
1119901
)
119903
)
1119903
le 119862
(5)
with the usualmodifications if119901 = infin or 119903 = infin Note that forparticular choices of 119904 119901 and 119903119861119904
119901119903(119872) contains the classical
Holder and Sobolev balls (see eg [40] and [19])
Remark 1 We have chosen a wavelet basis on [0 1] to fix thenotations wavelet basis on another interval can be consideredin the rest of the study without affecting the results
3 Statistical Framework Estimator and Result
31 Statistical Framework As mentioned in Section 1 anonparametric estimation setting as general as possible isadopted we aim to estimate an unknown function 119891 isin
L2([0 1]) via 119899 random variables (or vectors) 1198811 119881
119899from
a strictly stationary stochastic process (119881119905)119905isinZ defined on a
probability space (ΩAP) We suppose that (119881119905)119905isinZ has a
120572-mixing dependence structure with exponential decay ratethat is there exist two constants 120574 gt 0 and 120579 gt 0 such that
sup119898ge1
(119890120579119898120572119898) le 120574 (6)
where 120572119898
= sup(119860119861)isinF119881
minusinfin0timesF119881119898infin
|P(119860 cap 119861) minus P(119860)P(119861)|F119881
minusinfin0is the 120590-algebra generated by the random variables (or
vectors) 119881minus1
1198810and F119881
119898infinis the 120590-algebra generated by
the random variables (or vectors) 119881119898 119881
119898+1
The 120572-mixing dependence is reasonably weak it is sat-isfied by a wide variety of models including Markov chainsGARCH-type models and discretely observed diffusions(see for instance [1ndash3 41])
The considered estimator for 119891 is presented below
32 Estimator We define the hard thresholding waveletestimator 119891 by
119891 (119909) = sum119896isinΛ 1198950
1198881198950 119896
1206011198950 119896
(119909) +
1198951
sum119895=1198950
sum119896isinΛ 119895
1198891198951198961|119889119895119896|ge120581120582119895
120595119895119896
(119909)
(7)
where
119888119895119896
=1
119899
119899
sum119894=1
119902 (120601119895119896
119881119894) 119889
119895119896=
1
119899
119899
sum119894=1
119902 (120595119895119896
119881119894) (8)
Journal of Probability and Statistics 3
1 is the indicator function 120581 gt 0 is a large enough constant1198950is the integer satisfying
21198950 = [120591 ln 119899] (9)
where [119886] denotes the integer part of 119886 and 1198951is the integer
satisfying
21198951 = [(119899
(ln 119899)3)
1(2120588+1)
] (10)
120582119895= 2120588119895radic ln 119899
119899 (11)
Here it is supposed that there exists a function 119902 L2([0 1])times119881
1(Ω) rarr C such that
(H1) for 120574 isin 120601 120595 any integer 119895 ge 1198950and 119896 isin Λ
119895
E (119902 (120574119895119896
1198811)) = int
1
0
119891 (119909) 120574119895119896
(119909) 119889119909 (12)
where E denotes the expectation(H2) there exist two constants 119862 gt 0 and 120588 ge 0
satisfying for 120574 isin 120601 120595 for any integer 119895 ge 1198950
and 119896 isin Λ119895 (i) sup
119909isin1198811(Ω)|119902(120574
119895119896 119909)| le 119862212058811989521198952 (ii)
E(|119902(120574119895119896
1198811)|2) le 11986222120588119895 (iii) for any 119898 isin 1 119899 minus
1 ge 110038161003816100381610038161003816C119900V (119902 (120574
119895119896 119881
119898+1) 119902 (120574
119895119896 119881
1))
10038161003816100381610038161003816 le 119862221205881198952minus119895 (13)
whereC119900V denotes the covariance that isC119900V(119883 119884) =
E(119883119884)minusE(119883)E(119884) 119884 denotes the complex conjugateof 119884
For well-known nonparametric models in the iid set-ting hard thresholding wavelet estimators and importantresults can be found in for example Donoho and Johnstone[14 15] Donoho et al [16 17] Delyon and Juditsky [35]Kerkyacharian and Picard [36] and Fan and Koo [42] In the120572-mixing context 119891 defined by (7) is a general and improvedversion of the estimator considered in Chesneau [30 31] Themain differences are the presence of the tuning parameter 120588and the global definition of the function 119902 offering numerouspossibilities of applications Three of them are explored inSection 4
Comments on the Assumptions The assumption (H1) ensuresthat (8) are unbiased estimators for 119888
119895119896and 119889
119895119896given by
(4) whereas (H2) is related to their good performance SeeProposition 10These assumptions are not too restrictive Forinstance if we consider the standard density estimation prob-lem where (119881
119905)119905isinZ are iid random variables with bounded
density 119891 the function 119902(120574 119909) = 120574(119909) satisfies (H1) and (H2)with 120588 = 0 (note that thanks to the independence of (119881
119905)119905isinZ
the covariance term in (H2)-(iii) is zero)The technical detailsare given in Donoho et al [17]
Lemma 2 describes a simple situation in which assump-tion (H2)-(iii) is satisfied
Lemma 2 We make the following assumptions
(F1) Let119906 be the density of 1198811and let119906
(1198811 119881119898+1)be the density
of (1198811 119881
119898+1) for any 119898 isin Z We suppose that there
exists a constant 119862 gt 0 such that
sup119898isin1119899minus1Z
sup(119909119910)isin1198811(Ω)times119881119898+1(Ω)
10038161003816100381610038161003816119906(1198811 119881119898+1)(119909 119910)
minus119906 (119909) 119906 (119910)1003816100381610038161003816 le 119862
(14)
(F2) There exist two constants 119862 gt 0 and 120588 ge 0 satisfyingfor 120574 isin 120601 120595 for any integer 119895 ge 119895
0and 119896 isin Λ
119895
int1198811(Ω)
10038161003816100381610038161003816119902 (120574119895119896
119909)10038161003816100381610038161003816 119889119909 le 11986221205881198952minus1198952 (15)
Then under (F1) and (F2) (H2)-(iii) is satisfied
33 Result Theorem 3 determines the rate of convergenceattained by 119891 under the MISE over Besov balls
Theorem 3 We consider the general statistical settingdescribed in Section 31 Let 119891 be (7) under (H1) and (H2)Suppose that 119891 isin 119861119904
119901119903(119872) with 119903 ge 1 119901 ge 2 and 119904 isin (0119873)
or 119901 isin [1 2) and 119904 isin ((2120588 + 1)119901119873) Then there exists aconstant 119862 gt 0 such that
E (10038171003817100381710038171003817119891 minus 119891
100381710038171003817100381710038172
2) le 119862(
ln 119899
119899)
2119904(2119904+2120588+1)
(16)
The rate of convergence ldquo((ln 119899)119899)2119904(2119904+2120588+1)rdquo is oftenthe near optimal one in the minimax sense for numerousstatistical problems in a iid setting (see eg [19 43])Moreover note that Theorem 3 is flexible the assumptionson (119881
119905)119905isinZ related to the definition of 119902 in (H1) and (H2)
are mild In the next section this flexibility is illustrated forthree sophisticated nonparametric estimation problems thedensity deconvolution estimation problem the density esti-mation problem in a GARCH-typemodel and the regressionfunction estimation in the regression model with randomdesign
4 Applications
41 Density Deconvolution Let (119881119905)119905isinZ be a strictly stationary
stochastic process such that
119881119905= 119883
119905+ 120598
119905 119905 isin Z (17)
where (119883119905)119905isinZ is a strictly stationary stochastic process with
unknown density119891 and (120598119905)119905isinZ is a strictly stationary stochas-
tic process with known density 119892 It is supposed that 120598119905and
119883119905are independent for any 119905 isin Z and (119881
119905)119905isinZ is a 120572-mixing
process with exponential decay rate (see Section 31 for aprecise definition) Our aim is to estimate 119891 via 119881
1 119881
119899
from (119881119905)119905isinZ Some related works are Masry [44] Kulik [45]
Comte et al [46] and Van Zanten and Zareba [47]We formulate the following assumptions
4 Journal of Probability and Statistics
(G1) The support of 119891 is [0 1](G2) There exists a constant 119862 gt 0 such that
sup119909isinR
119891 (119909) le 119862 lt infin (18)
(G3) Let 119906 be the density of1198811We suppose that there exists
a constant 119862 gt 0 such that
sup119909isinR
119906 (119909) le 119862 (19)
(G4) For any 119898 isin Z let 119906(1198811 119881119898+1)
be the density of(119881
1 119881
119898+1) We suppose that there exists a constant
119862 gt 0 such that
sup119898isinZ
sup(119909119910)isinR2
119906(1198811 119881119898+1)
(119909 119910) le 119862 (20)
(G5) For any integrable function 120574 we define its Fouriertransform by
F (120574) (119909) = intinfin
minusinfin
120574 (119910) 119890minus119894119909119910119889119910 119909 isin R (21)
We suppose that there exist three known constants119862 gt 0119888 gt 0 and 120575 gt 1 such that for any 119909 isin R
(i) the Fourier transform of 119892 satisfies
1003816100381610038161003816F (119892) (119909)1003816100381610038161003816 ge
119888
(1 + 1199092)1205752
(22)
(ii) for any ℓ isin 0 1 2 the ℓth derivative of the Fouriertransform of 119892 satisfies
100381610038161003816100381610038161003816(F (119892) (119909))
(ℓ)100381610038161003816100381610038161003816le
119862
(1 + |119909|)120575+ℓ (23)
We are now in the position to present the result
Theorem 4 We consider the model (17) Suppose that (G1)ndash(G5) are satisfied Let 119891 be defined as in (7) with
119902 (120574 119909) =1
2120587int
infin
minusinfin
F (120574) (119910)
F (119892) (119910)119890minus119894119910119909119889119910 (24)
whereF(120574)(119910) denotes the complex conjugate ofF(120574)(119910) and120588 = 120575 (appearing in(G5))
Suppose that 119891 isin 119861119904
119901119903(119872) with 119903 ge 1 119901 ge 2 and 119904 isin
(0119873) or 119901 isin [1 2) and 119904 isin ((2120575 + 1)119901119873) Then thereexists a constant 119862 gt 0 such that
E (10038171003817100381710038171003817119891 minus 119891
100381710038171003817100381710038172
2) le 119862(
ln 119899
119899)
2119904(2119904+2120575+1)
(25)
Theorem 4 improves ([30] Proposition 51) in terms ofrate of convergence we gain a logarithmic term
Moreover it is established that in the iid settingldquo((ln 119899)119899)2119904(2119904+2120575+1)rdquo is
(i) exactly the rate of convergence attained by the hardthresholding wavelet estimator
(ii) the near optimal rate of convergence in the minimaxsense
The details can be found in Fan and Koo [42] ThusTheorem 4 can be viewed as an extension of this existingresult to the weak dependent case
42 GARCH-TypeModel We consider the strictly stationarystochastic process (119881
119905)119905isinZ where for any 119905 isin Z
119881119905= 1205902
119905119885
119905 (26)
(1205902
119905)119905isinZ is a strictly stationary stochastic process with
unknown density 119891 and (119885119905)119905isinZ is a strictly stationary
stochastic process with known density 119892 It is supposed that1205902
119905and 119885
119905are independent for any 119905 isin Z and (119881
119905)119905isinZ is a 120572-
mixing process with exponential decay rate (see Section 31for a precise definition) Our aim is to estimate 119891 via119881
1 119881
119899from (119881
119905)119905isinZ Some related works are Comte et al
[46] and Chesneau [31]We formulate the following assumptions(J1) There exists a positive integer 120575 such that
119892 (119909) =1
(120575 minus 1)(minus ln119909)
120575minus1 119909 isin [0 1] (27)
Let us remark that 119892 is the density of prod120575
119894=1119880
119894 where
1198801 119880
120575are 120575 iid random variables having the
common distributionU([0 1])(J2) The support of 119891 is [0 1] and 119891 isin L2([0 1])(J3) Let 119906 be the density of119881
1We suppose that there exists
a constant 119862 gt 0 such thatsup119909isinR
119906 (119909) le 119862 (28)
(J4) For any 119898 isin Z let 119906(1198811 119881119898+1)
be the density of(119881
1 119881
119898+1) We suppose that there exists a constant
119862 gt 0 such thatsup119898isinZ
sup(119909119910)isinR2
119906(1198811 119881119898+1)
(119909 119910) le 119862 (29)
We are now in the position to present the result
Theorem 5 We consider model (26) Suppose that (J1)ndash(J4)are satisfied Let 119891 be defined as in (7) with
119902 (120574 119909) = 119879120575(120574) (119909) (30)
where for any positive integer ℓ 119879(120574)(119909) = (119909120574(119909))1015840 and119879
ℓ(120574)(119909) = 119879(119879
ℓminus1(120574))(119909) and 120588 = 120575 (appearing in (J1))
Suppose that 119891 isin 119861119904
119901119903(119872) with 119903 ge 1 119901 ge 2 and 119904 isin
(0119873) or 119901 isin [1 2) and 119904 isin ((2120575 + 1)119901119873) Then thereexists a constant 119862 gt 0 such that
E (10038171003817100381710038171003817119891 minus 119891
100381710038171003817100381710038172
2) le 119862(
ln 119899
119899)
2119904(2119904+2120575+1)
(31)
Theorem 5 significantly improves ([31] Theorem 2) interms of rate of convergence we gain an exponent 12
Journal of Probability and Statistics 5
43 Nonparametric Regression Model We consider thestrictly stationary stochastic process (119881
119905)119905isinZ where for any
119905 isin Z 119881119905= (119884
119905 119883
119905)
119884119905= 119891 (119883
119905) + 120585
119905 (32)
(119883119905)119905isinZ is a strictly stationary stochastic process with
unknown density 119892 (120585119905)119905isinZ is a strictly stationary centered
stochastic process and119891 is the unknown regression functionIt is supposed that 119883
119905and 120585
119905are independent for any 119905 isin Z
and (119881119905)119905isinZ is a 120572-mixing process with exponential decay
rate (see Section 31 for a precise definition) Our aim is toestimate 119891 via 119881
1 119881
119899from (119881
119905)119905isinZ Applications of this
problem can be found in Hardle [48] Wavelet methods canbe found in Patil and Truong [23] Doosti et al [24] Doostiet al [26] and Doosti and Niroumand [25]
We formulate the following assumptions
(K1) The support of 119891 and 119892 is [0 1] and 119891 and 119892 isin
L2([0 1])(K2) 120585
1(Ω) is bounded
(K3) There exists a constant 119862 gt 0 such that
sup119909isin[01]
1003816100381610038161003816119891 (119909)1003816100381610038161003816 le 119862 (33)
(K4) There exist two constants 119888lowastgt 0 and 119862 gt 0 such that
119888lowastle inf
119909isin[01]
119892 (119909) sup119909isin[01]
119892 (119909) le 119862 (34)
(K5) Let 119906 be the density of1198811We suppose that there exists
a constant 119862 gt 0 such that
sup119909isinRtimes[01]
119906 (119909) le 119862 (35)
(K6) For any 119898 isin Z let 119906(1198811 119881119898+1)
be the density of(119881
1 119881
119898+1) We suppose that there exists a constant
119862 gt 0 such that
sup119898isinZ
sup(119909119910)isin(Rtimes[01])times(Rtimes[01])
119906(1198811 119881119898+1)
(119909 119910) le 119862 (36)
We are now in the position to present the result
Theorem 6 We consider the model (32) Suppose that (K1)ndash(K6) are satisfied Let 119891 be the truncated ratio estimatorConsider
119891 (119909) =V (119909)119892 (119909)
1|119892(119909)|ge119888lowast2
(37)
where
(i) V is defined as in (7) with
119902 (120574 (119909 119909lowast)) = 119909120574 (119909
lowast) (38)
and 120588 = 0
(ii) 119892 is defined as in (7) with 119883119905instead of 119881
119905
119902 (120574 119909) = 120574 (119909) (39)
and 120588 = 0(iii) 119888
lowastis the constant defined in (K4)
Suppose that 119891119892 isin 119861119904
119901119903(119872) and 119892 isin 119861119904
119901119903(119872) with 119903 ge 1
119901 ge 2 and 119904 isin (0119873) or 119901 isin [1 2) and 119904 isin (1119901119873) Thenthere exists a constant 119862 gt 0 such that
E (10038171003817100381710038171003817119891 minus 119891
100381710038171003817100381710038172
2) le 119862(
ln 119899
119899)
2119904(2119904+1)
(40)
The estimator (37) is derived by combining the procedureof Patil and Truong [23] with the truncated approach ofVasiliev [49]
Theorem 6 completes Patil and Truong [23] in terms ofrates of convergence under the MISE over Besov balls
Remark 7 The assumption(K2) can be relaxed with anotherstrategy to the one developed in Theorem 6 Some technicalelements are given in Chesneau [32]
Conclusion Considering the weak dependent case on theobservations we prove a general result on the rate of conver-gence attains by a hard wavelet thresholding estimator underthe MISE over Besov balls This result is flexible it can beapplied for a wide class of statistical models Moreover theobtained rate of convergence is sharp it can correspond to thenear optimal one in the minimax sense for the standard iidcase Some recent results on sophisticated statistical problemsare improved Thanks to its flexibility the perspectives ofapplications of our theoretical result in other contexts arenumerous
5 Proofs
In this section 119862 denotes any constant that does not dependon 119895 119896 and 119899 Its value may change from one term to anotherand may depend on 120601 or 120595
51 Key Lemmas Let us present two lemmas which will beused in the proofs
Lemma 8 shows a sharp covariance inequality under the120572-mixing condition
Lemma 8 (see [50]) Let (119882119905)119905isinZ be a strictly stationary 120572-
mixing process withmixing coefficient 120572119898119898 ge 0 and let ℎ and
119896 be two measurable functions Let 119901 gt 0 and 119902 gt 0 satisfying1119901 + 1119902 lt 1 such that E(|ℎ(119882
1)|119901) and E(|119896(119882
1)|119902) exist
Then there exists a constant 119862 gt 0 such that1003816100381610038161003816C119900V (ℎ (119882
1) 119896 (119882
119898+1))1003816100381610038161003816
le 1198621205721minus1119901minus1119902
119898(E (
1003816100381610038161003816ℎ (1198821)1003816100381610038161003816119901
))1119901
(E(1003816100381610038161003816119896 (119882
1)1003816100381610038161003816119902
))1119902
(41)
Lemma 9 below presents a concentration inequality for120572-mixing processes
6 Journal of Probability and Statistics
Lemma9 (see [13]) Let (119882119905)119905isinZ be a strictly stationary process
with the 119898th strongly mixing coefficient 120572119898 119898 ge 0 let 119899 be a
positive integer let ℎ R rarr C be a measurable function andfor any 119905 isin Z 119880
119905= ℎ(119882
119905) We assume that E(119880
1) = 0 and
there exists a constant 119872 gt 0 satisfying |1198801| le 119872 Then for
any 119898 isin 1 [1198992] and 120582 gt 0 we have
P(1
119899
100381610038161003816100381610038161003816100381610038161003816
119899
sum119894=1
119880119894
100381610038161003816100381610038161003816100381610038161003816ge 120582)
le 4 exp(minus1205822119899
16 (119863119898119898 + 1205821198721198983)
) + 32119872
120582119899120572
119898
(42)
where
119863119898
= max119897isin12119898
V (119897
sum119894=1
119880119894) (43)
52 Intermediary Results
Proof of Lemma 2 Using a standard expression of the covari-ance and (F1) as well as(F2) we obtain
10038161003816100381610038161003816C119900V (119902 (120574119895119896
119881119898+1
) 119902 (120574119895119896
1198811))
10038161003816100381610038161003816
=100381610038161003816100381610038161003816100381610038161003816int1198811(Ω)
int1198811(Ω)
119902 (120574119895119896
119909) 119902 (120574119895119896
119910)
times (119906(1198811 119881119898+1)
(119909 119910) minus 119906 (119909) 119906 (119910)) 119889119909119889119910100381610038161003816100381610038161003816100381610038161003816
le int1198811(Ω)
int1198811(Ω)
10038161003816100381610038161003816119902 (120574119895119896
119909)1003816100381610038161003816100381610038161003816100381610038161003816119902 (120574
119895119896 119910)
10038161003816100381610038161003816
times10038161003816100381610038161003816119906(1198811 119881119898+1)
(119909 119910) minus 119906 (119909) 119906 (119910)10038161003816100381610038161003816 119889119909 119889119910
le 119862(int1198811(Ω)
10038161003816100381610038161003816119902 (120574119895119896
119909)10038161003816100381610038161003816 119889119909)
2
le 119862221205881198952minus119895
(44)
This ends the proof of Lemma 2
Proposition 10 proves probability and moments inequal-ities satisfied by the estimators (8)
Proposition 10 Let 119895119896
and 120573119895119896
be defined as in (8) under(H1) and (H2) let 119895
0be (9) and let 119895
1be (10)
(a) There exists a constant 119862 gt 0 such that for any 119895 isin119895
0 119895
1 and 119896 isin Λ
119895
E (10038161003816100381610038161003816119888119895119896 minus 119888
119895119896
100381610038161003816100381610038162
) le 11986222120588119895 1
119899 (45)
E (10038161003816100381610038161003816119889119895119896
minus 119889119895119896
100381610038161003816100381610038162
) le 11986222120588119895 1
119899 (46)
(b) There exists a constant 119862 gt 0 such that for any 119895 isin119895
0 119895
1 and 119896 isin Λ
119895
E (10038161003816100381610038161003816119889119895119896
minus 119889119895119896
100381610038161003816100381610038164
) le 11986224120588119895 (47)
(c) Let 120582119895be defined as in (11) There exists a constant 119862 gt
0 such that for any 120581 large enough 119895 isin 1198950 119895
1 and
119896 isin Λ119895 we have
P(10038161003816100381610038161003816119889119895119896
minus 119889119895119896
10038161003816100381610038161003816 ge120581120582
119895
2) le 119862
1
1198994 (48)
Proof of Proposition 10 (a) Using (H1) and the stationarity of(119881
119905)119905isinZ we obtain
E (10038161003816100381610038161003816119888119895119896 minus 119888
119895119896
100381610038161003816100381610038162
) = V (119888119895119896
)
=1
1198992
119899
sum119894=1
V (119902 (120601119895119896
119881119894)) +
2
1198992
times119899
sumV=2
Vminus1
sumℓ=1
Re (C119900V (119902 (120601
119895119896 119881V) 119902 (120601
119895119896 119881
ℓ)))
=1
1198992
119899
sum119894=1
V (119902 (120601119895k 119881119894
)) +2
1198992
times119899minus1
sum119898=1
(119899 minus 119898)Re (C119900V (119902 (120601
119895119896 119881
119898+1) 119902 (120601
119895119896 119881
1)))
le1
119899(E (
10038161003816100381610038161003816119902 (120601119895119896
1198811)100381610038161003816100381610038162
)
+2119899minus1
sum119898=1
10038161003816100381610038161003816C119900V (119902 (120601119895119896
119881119898+1
) 119902 (120601119895119896
1198811))
10038161003816100381610038161003816)
(49)
By (H2)-(ii) we get
E (10038161003816100381610038161003816119902(120601119895119896
1198811)100381610038161003816100381610038162
) le 11986222120588119895 (50)
For the covariance term note that
119899minus1
sum119898=1
10038161003816100381610038161003816C119900V (119902 (120601119895119896
119881119898+1
) 119902 (120601119895119896
1198811))
10038161003816100381610038161003816 = 119860 + 119861 (51)
where
119860 =[ln 119899120579]minus1
sum119898=1
10038161003816100381610038161003816C119900V (119902 (120601119895119896
119881119898+1
) 119902 (120601119895119896
1198811))
10038161003816100381610038161003816
119861 =119899minus1
sum119898=[(ln 119899)120579]
10038161003816100381610038161003816C119900V (119902 (120601119895119896
119881119898+1
) 119902 (120601119895119896
1198811))
10038161003816100381610038161003816
(52)
It follows from (H2)-(iii) and 2minus119895 le 2minus1198950 lt 2(ln 119899)minus1 that
119860 le 119862221205881198952minus119895 [ln 119899
120579] le 11986222120588119895 (53)
Journal of Probability and Statistics 7
The Davydov inequality described in Lemma 8 with 119901 = 119902 =
4 (H2)-(i)-(ii) and 2119895 le 21198951 le 119899 give
119861 le 119862radicE (10038161003816100381610038161003816119902(120601119895119896
1198811)100381610038161003816100381610038164
)119899minus1
sum119898=[(ln 119899)120579]
radic120572119898
le 119862212058811989521198952radicE (10038161003816100381610038161003816119902(120601119895119896
1198811)100381610038161003816100381610038162
)infin
sum119898=[(ln 119899)120579]
119890minus1205791198982
= 11986222120588119895radic119899119890minus(ln 119899)2 le 11986222120588119895
(54)
Thus119899minus1
sum119898=1
10038161003816100381610038161003816C119900V (119902 (120601119895119896
119881119898+1
) 119902 (120601119895119896
1198811))
10038161003816100381610038161003816 le 11986222120588119895 (55)
Putting (49) (50) and (55) together the first point in (a)is proved The proof of the second point is identical with 120595instead of 120601
(b) Thanks to (H2)-(i) we have |119889119895119896
| le
sup119909isin1198811(Ω)
|119902(120595119895119896
119909)| le 119862212058811989521198952 It follows from thetriangular inequality and |119889
119895119896| le 119891
2le 119862 that
10038161003816100381610038161003816119889119895119896minus 119889
119895119896
10038161003816100381610038161003816 le10038161003816100381610038161003816119889119895119896
10038161003816100381610038161003816 +10038161003816100381610038161003816119889119895119896
10038161003816100381610038161003816 le 119862212058811989521198952 (56)
This inequality and the second result of (a) yield
E (10038161003816100381610038161003816119889119895119896
minus 119889119895119896
100381610038161003816100381610038164
) le 119862221205881198952119895E (
10038161003816100381610038161003816119889119895119896minus 119889
119895119896
100381610038161003816100381610038162
) le 119862241205881198952119895 1
119899
(57)
Using 2119895 le 21198951 le 119899 the proof of (b) is completed(c) We will use the Liebscher inequality described in
Lemma 9 Let us set
119880119894= 119902 (120595
119895119896 119881
119894) minus E (119902 (120595
119895119896 119881
1)) (58)
We have E(1198801) = 0 and by(H2)-(i) and 2119895 le 21198951 le
119899(ln 119899)3
1003816100381610038161003816119880119894
1003816100381610038161003816 le 2 sup119909isin1198811(Ω)
10038161003816100381610038161003816119902 (120595119895119896
119909)10038161003816100381610038161003816 le 119862212058811989521198952 le 1198622120588119895
radic119899
(ln 119899)3
(59)
(so 119872 = 1198622120588119895radic119899(ln 119899)3)Proceeding as for the proofs of the bounds in (a) for any
integer 119897 le 119862 ln 119899 since 2minus119895 le 2minus1198950 le 2(ln 119899)minus1 we show that
V (119897
sum119894=1
119880119894)
= V (119897
sum119894=1
119902 (120595119895119896
119881119894)) le 11986222120588119895 (119897 + 11989722minus119895) le 11986222120588119895119897
(60)
Therefore
119863119898
= max119897isin12119898
V (119897
sum119894=1
119880119894) le 11986222120588119895119898 (61)
Owing to Lemma 9 applied with 1198801 119880
119899 120582 = 120581120582
1198952
119898 = [radic120581 ln 119899] 119872 = 1198622120588119895radic119899(ln 119899)3 and the bound (61) weobtain
P(10038161003816100381610038161003816119889119895119896
minus 119889119895119896
10038161003816100381610038161003816 ge120581120582
119895
2)
le 119862(exp(minus11986212058121205822
119895119899
119863119898119898 + 120581120582
119895119898119872
) +119872
120582119895
119899119890minus120579119898)
le 119862( exp (minus119862((120581222120588119895 ln 119899)
times (22120588119895 + 1205812120588119895radic (ln 119899)
119899
times [radic120581 ln 119899] 2120588119895
radic119899
(ln 119899)3)
minus1
))
+ radic119899(ln 119899)3
(ln 119899) 119899119899119890minus120579[radic120581 ln 119899])
le 119862(119899minus1198621205812(1+120581
32) + 1198992minus120579radic120581)
(62)
Taking 120581 large enough the last term is bounded by1198621198994Thiscompletes the proof of(c)
This completes the proof of Proposition 10
Proof of Theorem 3 Theorem 3 can be proved by combiningarguments of ([36] Theorem 51) and ([51] Theorem 42) Itis close to ([30] Proof of Theorem 2) by taking 120579 rarr infin Theinterested reader can find the details below
We consider the following wavelet decomposition for 119891
119891 (119909) = sum119896isinΛ 1198950
1198881198950 119896
1206011198950 119896
(119909) +infin
sum119895=1198950
sum119896isinΛ 119895
119889119895119896
120595119895119896
(119909) (63)
where 1198881198950 119896
= int1
0119891(119909)120601
1198950 119896(119909)119889119909 and 119889
119895119896= int
1
0119891(119909)120595
119895119896(119909)119889119909
Using the orthonormality of the wavelet basis B theMISE of 119891 can be expressed as
E (10038171003817100381710038171003817119891 minus 119891
100381710038171003817100381710038172
2) = 119875 + 119876 + 119877 (64)
where
119875 = sum119896isinΛ 1198950
E (100381610038161003816100381610038161198881198950 119896 minus 119888
1198950 119896
100381610038161003816100381610038162
)
119876 =
1198951
sum119895=1198950
sum119896isinΛ 119895
E(100381610038161003816100381610038161003816119889
1198951198961|119889119895119896|ge120581120582119895
minus 119889119895119896
100381610038161003816100381610038161003816
2
)
119877 =infin
sum119895=1198951+1
sum119896isinΛ 119895
1198892
119895119896
(65)
8 Journal of Probability and Statistics
Let us now investigate sharp upper bounds for 119875 119877 and119876 successively
Upper Bound for 119875 The point (a) of Proposition 10 and2119904(2119904 + 2120588 + 1) lt 1 yield
119875 le 11986221198950
119899le 119862
ln 119899
119899le 119862(
ln 119899
119899)
2119904(2119904+2120588+1)
(66)
Upper Bound for 119877(i) For 119903 ge 1 and119901 ge 2 we have119891 isin 119861119904
119901119903(119872) sube 119861119904
2infin(119872)
Using 2119904(2119904 + 2120588 + 1) lt 2119904(2120588 + 1) we obtain
119877 le 119862infin
sum119895=1198951+1
2minus2119895119904 le 119862((ln 119899)3
119899)
2119904(2120588+1)
le 119862(ln 119899
119899)
2119904(2119904+2120588+1)
(67)
(ii) For 119903 ge 1 and 119901 isin [1 2) we have 119891 isin 119861119904
119901119903(119872) sube
119861119904+12minus1119901
2infin(119872) The condition 119904 gt (2120588 + 1)119901 implies
that (119904 + 12 minus 1119901)(2120588 + 1) gt 119904(2119904 + 2120588 + 1) Thus
119877 le 119862infin
sum119895=1198951+1
2minus2119895(119904+12minus1119901)
le 119862((ln 119899)3
119899)
2(119904+12minus1119901)(2120588+1)
le 119862(ln 119899
119899)
2119904(2119904+2120588+1)
(68)
Hence for 119903 ge 1 119901 ge 2 and 119904 gt 0 or 119901 isin [1 2) and119904 gt (2120588 + 1)119901 we have
119877 le 119862(ln 119899
119899)
2119904(2119904+2120588+1)
(69)
Upper Bound for 119876 Adopting the notation119863119895119896
= 119889119895119896
minus 119889119895119896
119876 can be written as
119876 =4
sum119894=1
119876119894 (70)
where
1198761=
1198951
sum119895=1198950
sum119896isinΛ 119895
E (10038161003816100381610038161003816119863119895119896
100381610038161003816100381610038162
1|119889119895119896|ge120581120582119895 |119889119895119896|lt1205811205821198952
)
1198762=
1198951
sum119895=1198950
sum119896isinΛ 119895
E (10038161003816100381610038161003816119863119895119896
100381610038161003816100381610038162
1|119889119895119896|ge120581120582119895 |119889119895119896|ge1205811205821198952
)
1198763=
1198951
sum119895=1198950
sum119896isinΛ 119895
E (1198892
1198951198961|119889119895119896|lt120581120582119895 |119889119895119896|ge2120581120582119895
)
1198764=
1198951
sum119895=1198950
sum119896isinΛ 119895
E (1198892
1198951198961|119889119895119896|lt120581120582119895 |119889119895119896|lt2120581120582119895
)
(71)
Upper Bound for 1198761
+ 1198763 Owing to the inequalities
1|119889119895119896|lt120581120582119895 |119889119895119896|ge2120581120582119895
le 1|119895119896|gt1205811205821198952
1|119889119895119896|ge120581120582119895 |119889119895119896|lt1205811205821198952
le
1|119895119896|gt1205811205821198952
and 1|119889119895119896|lt120581120582119895 |119889119895119896|ge2120581120582119895
le 1|119889119895119896|le2|119895119896|
theCauchy-Schwarz inequality and the points(b) and (c) ofProposition 10 we have
1198761+ 119876
3le 119862
1198951
sum119895=1198950
sum119896isinΛ 119895
E (10038161003816100381610038161003816119863119895119896
100381610038161003816100381610038162
1|119895119896|gt1205811205821198952
)
le 119862
1198951
sum119895=1198950
sum119896isinΛ 119895
(E (10038161003816100381610038161003816119863119895119896
100381610038161003816100381610038164
))12
(P(10038161003816100381610038161003816119863119895119896
10038161003816100381610038161003816 gt120581120582
119895
2))
12
le 1198621
1198992
1198951
sum119895=1198950
2119895(1+2120588) le 1198621
119899le 119862(
ln 119899
119899)
2119904(2119904+2120588+1)
(72)
Upper Bound for 1198762 It follows from the point(a) of
Proposition 10 that
1198762le
1198951
sum119895=1198950
sum119896isinΛ 119895
E (10038161003816100381610038161003816119863119895119896
100381610038161003816100381610038162
) 1|119889119895119896|ge1205811205821198952
le 1198621
119899
1198951
sum119895=j0
22120588119895 sum119896isinΛ 119895
1|119889119895119896|gt1205811205821198952
(73)
Let us now introduce the integer 119895lowastdefined by
2119895lowast = [(119899
ln 119899)
1(2119904+2120588+1)
] (74)
Note that 119895lowastisin 119895
0 119895
1 for 119899 large enough
Then 1198762can be bounded as
1198762le 119876
21+ 119876
22 (75)
where
11987621
= 1198621
119899
119895lowast
sum119895=1198950
22120588119895 sum119896isinΛ 119895
1|119889119895119896|gt1205811205821198952
11987622
= 1198621
119899
1198951
sum119895=119895lowast+1
22120588119895 sum119896isinΛ 119895
1|119889119895119896|gt1205811205821198952
(76)
On the one hand we have
11987621
le 119862ln 119899
119899
119895lowast
sum119895=1198950
2119895(1+2120588) le 119862(ln 119899
119899)
2119904(2119904+2120588+1)
(77)
On the other hand we have the following
Journal of Probability and Statistics 9
(i) For 119903 ge 1 and 119901 ge 2 the Markov inequality and 119891 isin119861119904
119901119903(119872) sube 119861119904
2infin(119872) yield
11987622
le 119862ln 119899
119899
1198951
sum119895=119895lowast+1
22120588119895 1
1205822
119895
sum119896isinΛ 119895
1198892
119895119896le 119862
infin
sum119895=119895lowast+1
sum119896isinΛ 119895
1198892
119895119896
le 119862infin
sum119895=119895lowast+1
2minus2119895119904 le 119862(ln 119899
119899)
2119904(2119904+2120588+1)
(78)
(ii) For 119903 ge 1 119901 isin [1 2) and 119904 gt (2120588 + 1)119901 the Markovinequality 119891 isin 119861119904
119901119903(119872) and (2119904 + 2120588 + 1)(2 minus 119901)2 +
(119904 + 12 minus 1119901 + 120588 minus 2120588119901)119901 = 2119904 imply that
11987622
le 119862ln 119899
119899
1198951
sum119895=119895lowast+1
22120588119895 1
120582119901
119895
sum119896isinΛ 119895
10038161003816100381610038161003816119889119895119896
10038161003816100381610038161003816119901
le 119862(ln 119899
119899)
(2minus119901)2 infin
sum119895=119895lowast+1
2119895120588(2minus119901)2minus119895(119904+12minus1119901)119901
le 119862(ln 119899
119899)
(2minus119901)2
2minus119895lowast(119904+12minus1119901+120588minus2120588119901)119901
le 119862(ln 119899
119899)
2119904(2119904+2120588+1)
(79)
Therefore for 119903 ge 1 119901 ge 2 and 119904 gt 0 or 119901 isin [1 2) and119904 gt (2120588 + 1)119901 we have
1198762le C(
ln 119899
119899)
2119904(2119904+2120588+1)
(80)
Upper Bound for 1198764 We have
1198764le
1198951
sum119895=1198950
sum119896isinΛ 119895
1198892
1198951198961|119889119895119896|lt2120581120582119895
(81)
Let 119895lowastbe the integer (74) Then 119876
4can be bound as
1198764le 119876
41+ 119876
42 (82)
where
11987641
=
119895lowast
sum119895=1198950
sum119896isinΛ 119895
1198892
1198951198961|119889119895119896|lt2120581120582119895
11987642
=
1198951
sum119895=119895lowast+1
sum119896isinΛ 119895
1198892
1198951198961|119889119895119896|lt2120581120582119895
(83)
On the one hand we have
11987641
le 119862
119895lowast
sum119895=1198950
21198951205822
119895= 119862
ln 119899
119899
119895lowast
sum119895=1198950
2119895(1+2120588) le 119862(ln 119899
119899)
2119904(2119904+2120588+1)
(84)
On the other hand we have the following
(i) For 119903 ge 1 and 119901 ge 2 since 119891 isin 119861119904
119901119903(119872) sube 119861119904
2infin(119872)
we have
11987642
leinfin
sum119895=119895lowast+1
sum119896isinΛ 119895
1198892
119895119896le 119862
infin
sum119895=119895lowast+1
2minus2119895119904 le 119862(ln 119899
119899)
2119904(2119904+2120588+1)
(85)
(ii) For 119903 ge 1 119901 isin [1 2) and 119904 gt (2120588 + 1)119901 owing to theMarkov inequality 119891 isin 119861119904
119901119903(119872) and (2119904 + 2120588 + 1)(2 minus
119901)2 + (119904 + 12 minus 1119901 + 120588 minus 2120588119901)119901 = 2119904 we get
11987642
le 119862
1198951
sum119895=119895lowast+1
1205822minus119901
119895sum
119896isinΛ 119895
10038161003816100381610038161003816119889119895119896
10038161003816100381610038161003816119901
= 119862(ln 119899
119899)
(2minus119901)2 1198951
sum119895=119895lowast+1
2119895120588(2minus119901) sum119896isinΛ 119895
10038161003816100381610038161003816119889119895119896
10038161003816100381610038161003816119901
le 119862(ln 119899
119899)
(2minus119901)2 infin
sum119895=119895lowast+1
2119895120588(2minus119901)2minus119895(119904+12minus1119901)119901
le 119862(ln 119899
119899)
(2minus119901)2
2minus119895lowast(119904+12minus1119901+120588minus2120588119901)119901
le 119862(ln 119899
119899)
2119904(2119904+2120588+1)
(86)
So for 119903 ge 1 119901 ge 2 and 119904 gt 0 or 119901 isin [1 2) and 119904 gt(2120588 + 1)119901 we have
1198764le 119862(
ln 119899
119899)
2119904(2119904+2120588+1)
(87)
Putting (70) (72) (80) and (87) together for 119903 ge 1 119901 ge 2and 119904 gt 0 or 119901 isin [1 2) and 119904 gt (2120588 + 1)119901 we obtain
119876 le 119862(ln 119899
119899)
2119904(2119904+2120588+1)
(88)
Combining (64) (66) (69) and (88) we complete theproof of Theorem 3
Proof of Theorem 4 The proof of Theorem 4 is a direct appli-cation ofTheorem 3 under (G1)ndash(G5) the function 119902 definedby (24) satisfies (H1) see ([42] equation (2)) and (H2) (i) see([42] Lemma 6) (ii) see ([42] equation (11)) and (iii) see([30] Proof of Proposition 61) with 120588 = 120575
Proof of Theorem 5 The proof ofTheorem 5 is a consequenceofTheorem 3 under (J1)ndash(J4) the function 119902 defined by (30)satisfies (H1) and (H2) (i)-(ii) see ([31] Proposition 1) and(iii) see ([52] equation (26)) with 120588 = 120575
Proof of Theorem 6 Set V(119909) = 119891(119909)119892(119909) Following themethodology of [49] we have
119891 (119909) minus 119891 (119909) = 119878 (119909) minus 119879 (119909) (89)
10 Journal of Probability and Statistics
where
119878 (119909)
=1
119892 (119909)(V (119909) minus V (119909)
+119891 (119909) (119892 (119909) minus 119892 (119909))) 1|119892(119909)|ge119888lowast2
119879 (119909) = 119891 (119909) 1|119892(119909)|lt119888lowast2
(90)
Using (K3) and the indicator function we have
|119878 (119909)| le 119862 (|V (119909) minus V (119909)| + 1003816100381610038161003816119892 (119909) minus 119892 (119909)1003816100381610038161003816) (91)
It follows from |119892(119909)| lt 119888lowast2 cap |119892(119909)| gt 119888
lowast sube |119892(119909) minus
119892(119909)| gt 119888lowast2 (K3) (K4) and the Markov inequality that
|119879 (119909)| le 1198621|119892(119909)minus119892(119909)|gt119888lowast2
le 1198621003816100381610038161003816119892 (119909) minus 119892 (119909)
1003816100381610038161003816 (92)
The triangular inequality yields
10038161003816100381610038161003816119891 (119909) minus 119891 (119909)10038161003816100381610038161003816 le 119862 (|V (119909) minus V (119909)| + 1003816100381610038161003816119892 (119909) minus 119892 (119909)
1003816100381610038161003816) (93)
The elementary inequality (119886 + 119887)2 le 2(1198862 + 1198872) implies that
E (10038171003817100381710038171003817119891 minus 119891
100381710038171003817100381710038172
2) le 119862 (E (V minus V2
2) + E (
1003817100381710038171003817119892 minus 11989210038171003817100381710038172
2)) (94)
We now bound this two MISEs via Theorem 3
Upper Bound for the MISE of V Under (K1)ndash(K6) thefunction 119902 defined by (38) satisfies the following
(H1) With V instead of 119891 since 1205851and 119883
1are independent
with E(1205851) = 0
E (119902 (120574119895119896
1198811))
= E (1198841120574119895119896
(1198831)) = E (119891 (119883
1) 120574
119895119896(119883
1))
= int1
0
119891 (119909) 120574119895119896
(119909) 119892 (119909) 119889119909 = int1
0
V (119909) 120574119895119896
(119909) 119889119909
(95)
(H2) (i)-(ii)-(iii) with 120588 = 0
(i) since 1198841(Ω) is bounded thanks to (K2) and (K3) say
|1198841| le 119872 with 119872 gt 0 we have
sup(119909119909lowast)isin1198811(Ω)
10038161003816100381610038161003816119902 (120574119895119896
(119909 119909lowast))
10038161003816100381610038161003816
= sup(119909119909lowast)isin[minus119872119872]times[01]
10038161003816100381610038161003816119909120574119895119896(119909
lowast)10038161003816100381610038161003816
le 119872 sup119909lowastisin[01]
10038161003816100381610038161003816120574119895119896(119909
lowast)10038161003816100381610038161003816 le 11986221198952
(96)
(ii) using the boundedness of 1198841(Ω) then (K4) we have
E (10038161003816100381610038161003816119902(120574119895119896
1198811)100381610038161003816100381610038162
) = E (1198842
1(120574
119895119896(119883
1))
2
)
le 119862E ((120574119895119896
(1198831))
2
)
= 119862int1
0
(120574119895119896
(119909))2
119892 (119909) 119889119909
le 119862int1
0
(120574119895119896
(119909))2
119889119909 le 119862
(97)
(iii) using the boundedness of 1198841(Ω) and making the
change of variables 119910 = 2119895119909 minus 119896 we obtain
int1198811(Ω)
10038161003816100381610038161003816119902 (120574119895119896
119909)10038161003816100381610038161003816 119889119909
= (int119872
minus119872
|119909| 119889119909)(int1
0
10038161003816100381610038161003816120574119895119896(119909
lowast)10038161003816100381610038161003816 119889119909lowast
)
= 1198722 int1
0
10038161003816100381610038161003816120574119895119896(119909)
10038161003816100381610038161003816 119889119909 le 1198622minus1198952
(98)
We conclude by applying Lemma 2 with 120588 = 0 (K5) and (K6)imply (F1) and the previous inequality implies (F2)
Therefore assuming that V isin 119861119904
119901119903(119872) with 119903 ge 1 119901 ge 2
and 119904 isin (0119873) or 119901 isin [1 2) and 119904 isin (1119901119873) Theorem 3proves the existence of a constant 119862 gt 0 satisfying
E (V minus V22) le 119862(
ln 119899
119899)
2119904(2119904+1)
(99)
Upper Bound for theMISE of 119892 Under (K1)ndash(K6) proceedingas the previous point we show that the function 119902 defined by(39) satisfies (H1) with 119892 instead of 119891 and 119883
119905instead of 119881
119905
and(H2) (i)-(ii)-(iii) with 120588 = 0Therefore assuming that 119892 isin 119861119904
119901119903(119872) with 119903 ge 1 119901 ge 2
and 119904 isin (0119873) or 119901 isin [1 2) and 119904 isin (1119901119873) Theorem 3proves the existance of a constant 119862 gt 0 satisfying
E (1003817100381710038171003817119892 minus 119892
10038171003817100381710038172
2) le 119862(
ln 119899
119899)
2119904(2119904+1)
(100)
Combining (94) (99) and (100) we end the proof ofTheorem 6
Conflict of Interests
The author declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
The author is thankful to the reviewers for their commentswhich have helped in improving the presentation
Journal of Probability and Statistics 11
References
[1] P DoukhanMixing Properties and Examples vol 85 of LectureNotes in Statistics Springer New York NY USA 1994
[2] M Carrasco and X Chen ldquoMixing and moment properties ofvarious GARCH and stochastic volatility modelsrdquo EconometricTheory vol 18 no 1 pp 17ndash39 2002
[3] R C Bradley Introduction to Strong Mixing Conditions Volume1 2 3 Kendrick Press 2007
[4] P M Robinson ldquoNonparametric estimators for time seriesrdquoJournal of Time Series Analysis vol 4 pp 185ndash207 1983
[5] G G Roussas ldquoNonparametric estimation inmixing sequencesof random variablesrdquo Journal of Statistical Planning and Infer-ence vol 18 no 2 pp 135ndash149 1987
[6] G G Roussas ldquoNonparametric regression estimation undermixing conditionsrdquo Stochastic Processes and their Applicationsvol 36 no 1 pp 107ndash116 1990
[7] Y K Truong and C J Stone ldquoNonparametric function estima-tion involving time seriesrdquo The Annals of Statistics vol 20 pp77ndash97 1992
[8] L H Tran ldquoNonparametric function estimation for time seriesby local average estimatorsrdquoThe Annals of Statistics vol 21 pp1040ndash1057 1993
[9] E Masry ldquoMultivariate local polynomial regression for timeseries uniform strong consistency and ratesrdquo Journal of TimeSeries Analysis vol 17 no 6 pp 571ndash599 1996
[10] E Masry ldquoMultivariate regression estimation local polynomialfitting for time seriesrdquo Stochastic Processes and their Applica-tions vol 65 no 1 pp 81ndash101 1996
[11] E Masry and J Fan ldquoLocal polynomial estimation of regressionfunctions for mixing processesrdquo Scandinavian Journal of Statis-tics vol 24 no 2 pp 165ndash179 1997
[12] D Bosq Nonparametric Statistics for Stochastic Processes Esti-mation and Prediction vol 110 of Lecture Notes in StatisticsSpringer New York NY USA 1998
[13] E Liebscher ldquoEstimation of the density and the regressionfunction under mixing conditionsrdquo Statistics and Decisions vol19 no 1 pp 9ndash26 2001
[14] D L Donoho and I M Johnstone ldquoIdeal spatial adaptation bywavelet shrinkagerdquo Biometrika vol 81 no 3 pp 425ndash455 1994
[15] D L Donoho and I M Johnstone ldquoAdapting to unknownsmoothness via wavelet shrinkagerdquo Journal of the AmericanStatistical Association vol 90 pp 1200ndash1224 1995
[16] D Donoho I Johnstone G Kerkyacharian and D PicardldquoWavelet shrinkage asymptopiardquo Journal of the Royal StatisticalSociety B vol 57 pp 301ndash369 1995
[17] D L Donoho I M Johnstone G Kerkyacharian and DPicard ldquoDensity estimation by wavelet thresholdingrdquo Annals ofStatistics vol 24 no 2 pp 508ndash539 1996
[18] A Antoniadis ldquoWavelets in statistics a review (with discus-sion)rdquo Journal of the Italian Statistical Society vol 6 pp 97ndash1441997
[19] W Hardle G Kerkyacharian D Picard and A TsybakovWavelet Approximation and Statistical Applications vol 129 ofLectures Notes in Statistics Springer New York NY USA 1998
[20] F Leblanc ldquoWavelet linear density estimator for a discrete-timestochastic process 119871
119901-lossesrdquo Statistics and Probability Letters
vol 27 no 1 pp 71ndash84 1996[21] K Tribouley andG Viennet ldquo119871
119901adaptive density estimation in
a alpha-mixing frameworkrdquo Annales de lrsquoinstitut Henri PoincareB vol 34 no 2 pp 179ndash208 1998
[22] E Masry ldquoWavelet-based estimation of multivariate regressionfunctions in besov spacesrdquo Journal of Nonparametric Statisticsvol 12 no 2 pp 283ndash308 2000
[23] P N Patil and Y K Truong ldquoAsymptotics for wavelet basedestimates of piecewise smooth regression for stationary timeseriesrdquo Annals of the Institute of Statistical Mathematics vol 53no 1 pp 159ndash178 2001
[24] H Doosti M Afshari and H A Niroumand ldquoWavelets fornonparametric stochastic regression with mixing stochasticprocessrdquo Communications in Statistics vol 37 no 3 pp 373ndash385 2008
[25] H Doosti and H A Niroumand ldquoMultivariate stochasticregression estimation by wavelet methods for stationary timeseriesrdquo Pakistan Journal of Statistics vol 27 no 1 pp 37ndash462009
[26] H Doosti M S Islam Y P Chaubey and P Gora ldquoTwo-dimensional wavelets for nonlinear autoregressive models withan application in dynamical systemrdquo Italian Journal of Pure andApplied Mathematics no 27 pp 39ndash62 2010
[27] J Cai and H Liang ldquoNonlinear wavelet density estimation fortruncated and dependent observationsrdquo International Journal ofWavelets Multiresolution and Information Processing vol 9 no4 pp 587ndash609 2011
[28] S Niu and H Liang ldquoNonlinear wavelet estimation of condi-tional density under left-truncated and 120572-mixing assumptionsrdquoInternational Journal of Wavelets Multiresolution and Informa-tion Processing vol 9 no 6 pp 989ndash1023 2011
[29] F Benatia and D Yahia ldquoNonlinear wavelet regression functionestimator for censored dependent datardquo Journal Afrika Statis-tika vol 7 no 1 pp 391ndash411 2012
[30] C Chesneau ldquoOn the adaptive wavelet deconvolution of adensity for strong mixing sequencesrdquo Journal of the KoreanStatistical Society vol 41 no 4 pp 423ndash436 2012
[31] C Chesneau ldquoWavelet estimation of a density in a GARCH-type modelrdquo Communications in Statistics vol 42 no 1 pp 98ndash117 2013
[32] C Chesneau ldquoOn the adaptive wavelet estimation of a mul-tidimensional regression function under alpha-mixing depen-dence beyond the standard assumptions on the noiserdquo Com-mentationes Mathematicae Universitatis Carolinae vol 4 pp527ndash556 2013
[33] Y P Chaubey and E Shirazi ldquoOn MISE of a nonlinear waveletestimator of the regression function based on biased data understrong mixing rdquo Communications in Statistics In press
[34] M Abbaszadeh andM Emadi ldquoWavelet density estimation andstatistical evidences role for a garch model in the weighteddistributionrdquo Applied Mathematics vol 4 no 2 pp 10ndash4162013
[35] B Delyon and A Juditsky ldquoOn minimax wavelet estimatorsrdquoApplied and Computational Harmonic Analysis vol 3 no 3 pp215ndash228 1996
[36] G Kerkyacharian and D Picard ldquoThresholding algorithmsmaxisets and well-concentrated basesrdquo Test vol 9 no 2 pp283ndash344 2000
[37] I Daubechies Ten Lectures on Wavelets SIAM 1992[38] A Cohen I Daubechies and P Vial ldquoWavelets on the interval
and fast wavelet transformsrdquo Applied and Computational Har-monic Analysis vol 1 no 1 pp 54ndash81 1993
[39] S Mallat A Wavelet Tour of Signal Processing The sparse waywith contributions fromGabriel Peyralphae ElsevierAcademicPress Amsterdam The Netherlands 3rd edition 2009
12 Journal of Probability and Statistics
[40] Y Meyer Ondelettes et Opalphaerateurs Hermann ParisFrance 1990
[41] V Genon-Catalot T Jeantheau and C Laredo ldquoStochasticvolatility models as hidden Markov models and statisticalapplicationsrdquo Bernoulli vol 6 no 6 pp 1051ndash1079 2000
[42] J Fan and J Koo ldquoWavelet deconvolutionrdquo IEEE Transactionson Information Theory vol 48 no 3 pp 734ndash747 2002
[43] A B Tsybakov Introduction alphaa lrsquoestimation non-paramalphaetrique Springer New York NY USA 2004
[44] E Masry ldquoStrong consistency and rates for deconvolutionof multivariate densities of stationary processesrdquo StochasticProcesses and their Applications vol 47 no 1 pp 53ndash74 1993
[45] R Kulik ldquoNonparametric deconvolution problem for depen-dent sequencesrdquo Electronic Journal of Statistics vol 2 pp 722ndash740 2008
[46] F Comte J Dedecker and M L Taupin ldquoAdaptive densitydeconvolution with dependent inputsrdquo Mathematical Methodsof Statistics vol 17 no 2 pp 87ndash112 2008
[47] H van Zanten and P Zareba ldquoA note on wavelet densitydeconvolution for weakly dependent datardquo Statistical Inferencefor Stochastic Processes vol 11 no 2 pp 207ndash219 2008
[48] W Hardle Applied Nonparametric Regression Cambridge Uni-versity Press Cambridge UK 1990
[49] V A Vasiliev ldquoOne investigation method of a ratios typeestimatorsrdquo in Proceedings of the 16th IFAC Symposium onSystem Identialphacation pp 1ndash6 Brussels Belgium July 2012(in progress)
[50] Y Davydov ldquoThe invariance principle for stationary processesrdquoTheory of Probability and Its Applications vol 15 no 3 pp 498ndash509 1970
[51] C Chesneau ldquoWavelet estimation via block thresholding aminimax study under 119871119901-riskrdquo Statistica Sinica vol 18 no 3pp 1007ndash1024 2008
[52] C Chesneau and H Doosti ldquoWavelet linear density estimationfor a GARCH model under various dependence structuresrdquoJournal of the Iranian Statistical Society vol 11 no 1 pp 1ndash212012
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Journal of Probability and Statistics 3
1 is the indicator function 120581 gt 0 is a large enough constant1198950is the integer satisfying
21198950 = [120591 ln 119899] (9)
where [119886] denotes the integer part of 119886 and 1198951is the integer
satisfying
21198951 = [(119899
(ln 119899)3)
1(2120588+1)
] (10)
120582119895= 2120588119895radic ln 119899
119899 (11)
Here it is supposed that there exists a function 119902 L2([0 1])times119881
1(Ω) rarr C such that
(H1) for 120574 isin 120601 120595 any integer 119895 ge 1198950and 119896 isin Λ
119895
E (119902 (120574119895119896
1198811)) = int
1
0
119891 (119909) 120574119895119896
(119909) 119889119909 (12)
where E denotes the expectation(H2) there exist two constants 119862 gt 0 and 120588 ge 0
satisfying for 120574 isin 120601 120595 for any integer 119895 ge 1198950
and 119896 isin Λ119895 (i) sup
119909isin1198811(Ω)|119902(120574
119895119896 119909)| le 119862212058811989521198952 (ii)
E(|119902(120574119895119896
1198811)|2) le 11986222120588119895 (iii) for any 119898 isin 1 119899 minus
1 ge 110038161003816100381610038161003816C119900V (119902 (120574
119895119896 119881
119898+1) 119902 (120574
119895119896 119881
1))
10038161003816100381610038161003816 le 119862221205881198952minus119895 (13)
whereC119900V denotes the covariance that isC119900V(119883 119884) =
E(119883119884)minusE(119883)E(119884) 119884 denotes the complex conjugateof 119884
For well-known nonparametric models in the iid set-ting hard thresholding wavelet estimators and importantresults can be found in for example Donoho and Johnstone[14 15] Donoho et al [16 17] Delyon and Juditsky [35]Kerkyacharian and Picard [36] and Fan and Koo [42] In the120572-mixing context 119891 defined by (7) is a general and improvedversion of the estimator considered in Chesneau [30 31] Themain differences are the presence of the tuning parameter 120588and the global definition of the function 119902 offering numerouspossibilities of applications Three of them are explored inSection 4
Comments on the Assumptions The assumption (H1) ensuresthat (8) are unbiased estimators for 119888
119895119896and 119889
119895119896given by
(4) whereas (H2) is related to their good performance SeeProposition 10These assumptions are not too restrictive Forinstance if we consider the standard density estimation prob-lem where (119881
119905)119905isinZ are iid random variables with bounded
density 119891 the function 119902(120574 119909) = 120574(119909) satisfies (H1) and (H2)with 120588 = 0 (note that thanks to the independence of (119881
119905)119905isinZ
the covariance term in (H2)-(iii) is zero)The technical detailsare given in Donoho et al [17]
Lemma 2 describes a simple situation in which assump-tion (H2)-(iii) is satisfied
Lemma 2 We make the following assumptions
(F1) Let119906 be the density of 1198811and let119906
(1198811 119881119898+1)be the density
of (1198811 119881
119898+1) for any 119898 isin Z We suppose that there
exists a constant 119862 gt 0 such that
sup119898isin1119899minus1Z
sup(119909119910)isin1198811(Ω)times119881119898+1(Ω)
10038161003816100381610038161003816119906(1198811 119881119898+1)(119909 119910)
minus119906 (119909) 119906 (119910)1003816100381610038161003816 le 119862
(14)
(F2) There exist two constants 119862 gt 0 and 120588 ge 0 satisfyingfor 120574 isin 120601 120595 for any integer 119895 ge 119895
0and 119896 isin Λ
119895
int1198811(Ω)
10038161003816100381610038161003816119902 (120574119895119896
119909)10038161003816100381610038161003816 119889119909 le 11986221205881198952minus1198952 (15)
Then under (F1) and (F2) (H2)-(iii) is satisfied
33 Result Theorem 3 determines the rate of convergenceattained by 119891 under the MISE over Besov balls
Theorem 3 We consider the general statistical settingdescribed in Section 31 Let 119891 be (7) under (H1) and (H2)Suppose that 119891 isin 119861119904
119901119903(119872) with 119903 ge 1 119901 ge 2 and 119904 isin (0119873)
or 119901 isin [1 2) and 119904 isin ((2120588 + 1)119901119873) Then there exists aconstant 119862 gt 0 such that
E (10038171003817100381710038171003817119891 minus 119891
100381710038171003817100381710038172
2) le 119862(
ln 119899
119899)
2119904(2119904+2120588+1)
(16)
The rate of convergence ldquo((ln 119899)119899)2119904(2119904+2120588+1)rdquo is oftenthe near optimal one in the minimax sense for numerousstatistical problems in a iid setting (see eg [19 43])Moreover note that Theorem 3 is flexible the assumptionson (119881
119905)119905isinZ related to the definition of 119902 in (H1) and (H2)
are mild In the next section this flexibility is illustrated forthree sophisticated nonparametric estimation problems thedensity deconvolution estimation problem the density esti-mation problem in a GARCH-typemodel and the regressionfunction estimation in the regression model with randomdesign
4 Applications
41 Density Deconvolution Let (119881119905)119905isinZ be a strictly stationary
stochastic process such that
119881119905= 119883
119905+ 120598
119905 119905 isin Z (17)
where (119883119905)119905isinZ is a strictly stationary stochastic process with
unknown density119891 and (120598119905)119905isinZ is a strictly stationary stochas-
tic process with known density 119892 It is supposed that 120598119905and
119883119905are independent for any 119905 isin Z and (119881
119905)119905isinZ is a 120572-mixing
process with exponential decay rate (see Section 31 for aprecise definition) Our aim is to estimate 119891 via 119881
1 119881
119899
from (119881119905)119905isinZ Some related works are Masry [44] Kulik [45]
Comte et al [46] and Van Zanten and Zareba [47]We formulate the following assumptions
4 Journal of Probability and Statistics
(G1) The support of 119891 is [0 1](G2) There exists a constant 119862 gt 0 such that
sup119909isinR
119891 (119909) le 119862 lt infin (18)
(G3) Let 119906 be the density of1198811We suppose that there exists
a constant 119862 gt 0 such that
sup119909isinR
119906 (119909) le 119862 (19)
(G4) For any 119898 isin Z let 119906(1198811 119881119898+1)
be the density of(119881
1 119881
119898+1) We suppose that there exists a constant
119862 gt 0 such that
sup119898isinZ
sup(119909119910)isinR2
119906(1198811 119881119898+1)
(119909 119910) le 119862 (20)
(G5) For any integrable function 120574 we define its Fouriertransform by
F (120574) (119909) = intinfin
minusinfin
120574 (119910) 119890minus119894119909119910119889119910 119909 isin R (21)
We suppose that there exist three known constants119862 gt 0119888 gt 0 and 120575 gt 1 such that for any 119909 isin R
(i) the Fourier transform of 119892 satisfies
1003816100381610038161003816F (119892) (119909)1003816100381610038161003816 ge
119888
(1 + 1199092)1205752
(22)
(ii) for any ℓ isin 0 1 2 the ℓth derivative of the Fouriertransform of 119892 satisfies
100381610038161003816100381610038161003816(F (119892) (119909))
(ℓ)100381610038161003816100381610038161003816le
119862
(1 + |119909|)120575+ℓ (23)
We are now in the position to present the result
Theorem 4 We consider the model (17) Suppose that (G1)ndash(G5) are satisfied Let 119891 be defined as in (7) with
119902 (120574 119909) =1
2120587int
infin
minusinfin
F (120574) (119910)
F (119892) (119910)119890minus119894119910119909119889119910 (24)
whereF(120574)(119910) denotes the complex conjugate ofF(120574)(119910) and120588 = 120575 (appearing in(G5))
Suppose that 119891 isin 119861119904
119901119903(119872) with 119903 ge 1 119901 ge 2 and 119904 isin
(0119873) or 119901 isin [1 2) and 119904 isin ((2120575 + 1)119901119873) Then thereexists a constant 119862 gt 0 such that
E (10038171003817100381710038171003817119891 minus 119891
100381710038171003817100381710038172
2) le 119862(
ln 119899
119899)
2119904(2119904+2120575+1)
(25)
Theorem 4 improves ([30] Proposition 51) in terms ofrate of convergence we gain a logarithmic term
Moreover it is established that in the iid settingldquo((ln 119899)119899)2119904(2119904+2120575+1)rdquo is
(i) exactly the rate of convergence attained by the hardthresholding wavelet estimator
(ii) the near optimal rate of convergence in the minimaxsense
The details can be found in Fan and Koo [42] ThusTheorem 4 can be viewed as an extension of this existingresult to the weak dependent case
42 GARCH-TypeModel We consider the strictly stationarystochastic process (119881
119905)119905isinZ where for any 119905 isin Z
119881119905= 1205902
119905119885
119905 (26)
(1205902
119905)119905isinZ is a strictly stationary stochastic process with
unknown density 119891 and (119885119905)119905isinZ is a strictly stationary
stochastic process with known density 119892 It is supposed that1205902
119905and 119885
119905are independent for any 119905 isin Z and (119881
119905)119905isinZ is a 120572-
mixing process with exponential decay rate (see Section 31for a precise definition) Our aim is to estimate 119891 via119881
1 119881
119899from (119881
119905)119905isinZ Some related works are Comte et al
[46] and Chesneau [31]We formulate the following assumptions(J1) There exists a positive integer 120575 such that
119892 (119909) =1
(120575 minus 1)(minus ln119909)
120575minus1 119909 isin [0 1] (27)
Let us remark that 119892 is the density of prod120575
119894=1119880
119894 where
1198801 119880
120575are 120575 iid random variables having the
common distributionU([0 1])(J2) The support of 119891 is [0 1] and 119891 isin L2([0 1])(J3) Let 119906 be the density of119881
1We suppose that there exists
a constant 119862 gt 0 such thatsup119909isinR
119906 (119909) le 119862 (28)
(J4) For any 119898 isin Z let 119906(1198811 119881119898+1)
be the density of(119881
1 119881
119898+1) We suppose that there exists a constant
119862 gt 0 such thatsup119898isinZ
sup(119909119910)isinR2
119906(1198811 119881119898+1)
(119909 119910) le 119862 (29)
We are now in the position to present the result
Theorem 5 We consider model (26) Suppose that (J1)ndash(J4)are satisfied Let 119891 be defined as in (7) with
119902 (120574 119909) = 119879120575(120574) (119909) (30)
where for any positive integer ℓ 119879(120574)(119909) = (119909120574(119909))1015840 and119879
ℓ(120574)(119909) = 119879(119879
ℓminus1(120574))(119909) and 120588 = 120575 (appearing in (J1))
Suppose that 119891 isin 119861119904
119901119903(119872) with 119903 ge 1 119901 ge 2 and 119904 isin
(0119873) or 119901 isin [1 2) and 119904 isin ((2120575 + 1)119901119873) Then thereexists a constant 119862 gt 0 such that
E (10038171003817100381710038171003817119891 minus 119891
100381710038171003817100381710038172
2) le 119862(
ln 119899
119899)
2119904(2119904+2120575+1)
(31)
Theorem 5 significantly improves ([31] Theorem 2) interms of rate of convergence we gain an exponent 12
Journal of Probability and Statistics 5
43 Nonparametric Regression Model We consider thestrictly stationary stochastic process (119881
119905)119905isinZ where for any
119905 isin Z 119881119905= (119884
119905 119883
119905)
119884119905= 119891 (119883
119905) + 120585
119905 (32)
(119883119905)119905isinZ is a strictly stationary stochastic process with
unknown density 119892 (120585119905)119905isinZ is a strictly stationary centered
stochastic process and119891 is the unknown regression functionIt is supposed that 119883
119905and 120585
119905are independent for any 119905 isin Z
and (119881119905)119905isinZ is a 120572-mixing process with exponential decay
rate (see Section 31 for a precise definition) Our aim is toestimate 119891 via 119881
1 119881
119899from (119881
119905)119905isinZ Applications of this
problem can be found in Hardle [48] Wavelet methods canbe found in Patil and Truong [23] Doosti et al [24] Doostiet al [26] and Doosti and Niroumand [25]
We formulate the following assumptions
(K1) The support of 119891 and 119892 is [0 1] and 119891 and 119892 isin
L2([0 1])(K2) 120585
1(Ω) is bounded
(K3) There exists a constant 119862 gt 0 such that
sup119909isin[01]
1003816100381610038161003816119891 (119909)1003816100381610038161003816 le 119862 (33)
(K4) There exist two constants 119888lowastgt 0 and 119862 gt 0 such that
119888lowastle inf
119909isin[01]
119892 (119909) sup119909isin[01]
119892 (119909) le 119862 (34)
(K5) Let 119906 be the density of1198811We suppose that there exists
a constant 119862 gt 0 such that
sup119909isinRtimes[01]
119906 (119909) le 119862 (35)
(K6) For any 119898 isin Z let 119906(1198811 119881119898+1)
be the density of(119881
1 119881
119898+1) We suppose that there exists a constant
119862 gt 0 such that
sup119898isinZ
sup(119909119910)isin(Rtimes[01])times(Rtimes[01])
119906(1198811 119881119898+1)
(119909 119910) le 119862 (36)
We are now in the position to present the result
Theorem 6 We consider the model (32) Suppose that (K1)ndash(K6) are satisfied Let 119891 be the truncated ratio estimatorConsider
119891 (119909) =V (119909)119892 (119909)
1|119892(119909)|ge119888lowast2
(37)
where
(i) V is defined as in (7) with
119902 (120574 (119909 119909lowast)) = 119909120574 (119909
lowast) (38)
and 120588 = 0
(ii) 119892 is defined as in (7) with 119883119905instead of 119881
119905
119902 (120574 119909) = 120574 (119909) (39)
and 120588 = 0(iii) 119888
lowastis the constant defined in (K4)
Suppose that 119891119892 isin 119861119904
119901119903(119872) and 119892 isin 119861119904
119901119903(119872) with 119903 ge 1
119901 ge 2 and 119904 isin (0119873) or 119901 isin [1 2) and 119904 isin (1119901119873) Thenthere exists a constant 119862 gt 0 such that
E (10038171003817100381710038171003817119891 minus 119891
100381710038171003817100381710038172
2) le 119862(
ln 119899
119899)
2119904(2119904+1)
(40)
The estimator (37) is derived by combining the procedureof Patil and Truong [23] with the truncated approach ofVasiliev [49]
Theorem 6 completes Patil and Truong [23] in terms ofrates of convergence under the MISE over Besov balls
Remark 7 The assumption(K2) can be relaxed with anotherstrategy to the one developed in Theorem 6 Some technicalelements are given in Chesneau [32]
Conclusion Considering the weak dependent case on theobservations we prove a general result on the rate of conver-gence attains by a hard wavelet thresholding estimator underthe MISE over Besov balls This result is flexible it can beapplied for a wide class of statistical models Moreover theobtained rate of convergence is sharp it can correspond to thenear optimal one in the minimax sense for the standard iidcase Some recent results on sophisticated statistical problemsare improved Thanks to its flexibility the perspectives ofapplications of our theoretical result in other contexts arenumerous
5 Proofs
In this section 119862 denotes any constant that does not dependon 119895 119896 and 119899 Its value may change from one term to anotherand may depend on 120601 or 120595
51 Key Lemmas Let us present two lemmas which will beused in the proofs
Lemma 8 shows a sharp covariance inequality under the120572-mixing condition
Lemma 8 (see [50]) Let (119882119905)119905isinZ be a strictly stationary 120572-
mixing process withmixing coefficient 120572119898119898 ge 0 and let ℎ and
119896 be two measurable functions Let 119901 gt 0 and 119902 gt 0 satisfying1119901 + 1119902 lt 1 such that E(|ℎ(119882
1)|119901) and E(|119896(119882
1)|119902) exist
Then there exists a constant 119862 gt 0 such that1003816100381610038161003816C119900V (ℎ (119882
1) 119896 (119882
119898+1))1003816100381610038161003816
le 1198621205721minus1119901minus1119902
119898(E (
1003816100381610038161003816ℎ (1198821)1003816100381610038161003816119901
))1119901
(E(1003816100381610038161003816119896 (119882
1)1003816100381610038161003816119902
))1119902
(41)
Lemma 9 below presents a concentration inequality for120572-mixing processes
6 Journal of Probability and Statistics
Lemma9 (see [13]) Let (119882119905)119905isinZ be a strictly stationary process
with the 119898th strongly mixing coefficient 120572119898 119898 ge 0 let 119899 be a
positive integer let ℎ R rarr C be a measurable function andfor any 119905 isin Z 119880
119905= ℎ(119882
119905) We assume that E(119880
1) = 0 and
there exists a constant 119872 gt 0 satisfying |1198801| le 119872 Then for
any 119898 isin 1 [1198992] and 120582 gt 0 we have
P(1
119899
100381610038161003816100381610038161003816100381610038161003816
119899
sum119894=1
119880119894
100381610038161003816100381610038161003816100381610038161003816ge 120582)
le 4 exp(minus1205822119899
16 (119863119898119898 + 1205821198721198983)
) + 32119872
120582119899120572
119898
(42)
where
119863119898
= max119897isin12119898
V (119897
sum119894=1
119880119894) (43)
52 Intermediary Results
Proof of Lemma 2 Using a standard expression of the covari-ance and (F1) as well as(F2) we obtain
10038161003816100381610038161003816C119900V (119902 (120574119895119896
119881119898+1
) 119902 (120574119895119896
1198811))
10038161003816100381610038161003816
=100381610038161003816100381610038161003816100381610038161003816int1198811(Ω)
int1198811(Ω)
119902 (120574119895119896
119909) 119902 (120574119895119896
119910)
times (119906(1198811 119881119898+1)
(119909 119910) minus 119906 (119909) 119906 (119910)) 119889119909119889119910100381610038161003816100381610038161003816100381610038161003816
le int1198811(Ω)
int1198811(Ω)
10038161003816100381610038161003816119902 (120574119895119896
119909)1003816100381610038161003816100381610038161003816100381610038161003816119902 (120574
119895119896 119910)
10038161003816100381610038161003816
times10038161003816100381610038161003816119906(1198811 119881119898+1)
(119909 119910) minus 119906 (119909) 119906 (119910)10038161003816100381610038161003816 119889119909 119889119910
le 119862(int1198811(Ω)
10038161003816100381610038161003816119902 (120574119895119896
119909)10038161003816100381610038161003816 119889119909)
2
le 119862221205881198952minus119895
(44)
This ends the proof of Lemma 2
Proposition 10 proves probability and moments inequal-ities satisfied by the estimators (8)
Proposition 10 Let 119895119896
and 120573119895119896
be defined as in (8) under(H1) and (H2) let 119895
0be (9) and let 119895
1be (10)
(a) There exists a constant 119862 gt 0 such that for any 119895 isin119895
0 119895
1 and 119896 isin Λ
119895
E (10038161003816100381610038161003816119888119895119896 minus 119888
119895119896
100381610038161003816100381610038162
) le 11986222120588119895 1
119899 (45)
E (10038161003816100381610038161003816119889119895119896
minus 119889119895119896
100381610038161003816100381610038162
) le 11986222120588119895 1
119899 (46)
(b) There exists a constant 119862 gt 0 such that for any 119895 isin119895
0 119895
1 and 119896 isin Λ
119895
E (10038161003816100381610038161003816119889119895119896
minus 119889119895119896
100381610038161003816100381610038164
) le 11986224120588119895 (47)
(c) Let 120582119895be defined as in (11) There exists a constant 119862 gt
0 such that for any 120581 large enough 119895 isin 1198950 119895
1 and
119896 isin Λ119895 we have
P(10038161003816100381610038161003816119889119895119896
minus 119889119895119896
10038161003816100381610038161003816 ge120581120582
119895
2) le 119862
1
1198994 (48)
Proof of Proposition 10 (a) Using (H1) and the stationarity of(119881
119905)119905isinZ we obtain
E (10038161003816100381610038161003816119888119895119896 minus 119888
119895119896
100381610038161003816100381610038162
) = V (119888119895119896
)
=1
1198992
119899
sum119894=1
V (119902 (120601119895119896
119881119894)) +
2
1198992
times119899
sumV=2
Vminus1
sumℓ=1
Re (C119900V (119902 (120601
119895119896 119881V) 119902 (120601
119895119896 119881
ℓ)))
=1
1198992
119899
sum119894=1
V (119902 (120601119895k 119881119894
)) +2
1198992
times119899minus1
sum119898=1
(119899 minus 119898)Re (C119900V (119902 (120601
119895119896 119881
119898+1) 119902 (120601
119895119896 119881
1)))
le1
119899(E (
10038161003816100381610038161003816119902 (120601119895119896
1198811)100381610038161003816100381610038162
)
+2119899minus1
sum119898=1
10038161003816100381610038161003816C119900V (119902 (120601119895119896
119881119898+1
) 119902 (120601119895119896
1198811))
10038161003816100381610038161003816)
(49)
By (H2)-(ii) we get
E (10038161003816100381610038161003816119902(120601119895119896
1198811)100381610038161003816100381610038162
) le 11986222120588119895 (50)
For the covariance term note that
119899minus1
sum119898=1
10038161003816100381610038161003816C119900V (119902 (120601119895119896
119881119898+1
) 119902 (120601119895119896
1198811))
10038161003816100381610038161003816 = 119860 + 119861 (51)
where
119860 =[ln 119899120579]minus1
sum119898=1
10038161003816100381610038161003816C119900V (119902 (120601119895119896
119881119898+1
) 119902 (120601119895119896
1198811))
10038161003816100381610038161003816
119861 =119899minus1
sum119898=[(ln 119899)120579]
10038161003816100381610038161003816C119900V (119902 (120601119895119896
119881119898+1
) 119902 (120601119895119896
1198811))
10038161003816100381610038161003816
(52)
It follows from (H2)-(iii) and 2minus119895 le 2minus1198950 lt 2(ln 119899)minus1 that
119860 le 119862221205881198952minus119895 [ln 119899
120579] le 11986222120588119895 (53)
Journal of Probability and Statistics 7
The Davydov inequality described in Lemma 8 with 119901 = 119902 =
4 (H2)-(i)-(ii) and 2119895 le 21198951 le 119899 give
119861 le 119862radicE (10038161003816100381610038161003816119902(120601119895119896
1198811)100381610038161003816100381610038164
)119899minus1
sum119898=[(ln 119899)120579]
radic120572119898
le 119862212058811989521198952radicE (10038161003816100381610038161003816119902(120601119895119896
1198811)100381610038161003816100381610038162
)infin
sum119898=[(ln 119899)120579]
119890minus1205791198982
= 11986222120588119895radic119899119890minus(ln 119899)2 le 11986222120588119895
(54)
Thus119899minus1
sum119898=1
10038161003816100381610038161003816C119900V (119902 (120601119895119896
119881119898+1
) 119902 (120601119895119896
1198811))
10038161003816100381610038161003816 le 11986222120588119895 (55)
Putting (49) (50) and (55) together the first point in (a)is proved The proof of the second point is identical with 120595instead of 120601
(b) Thanks to (H2)-(i) we have |119889119895119896
| le
sup119909isin1198811(Ω)
|119902(120595119895119896
119909)| le 119862212058811989521198952 It follows from thetriangular inequality and |119889
119895119896| le 119891
2le 119862 that
10038161003816100381610038161003816119889119895119896minus 119889
119895119896
10038161003816100381610038161003816 le10038161003816100381610038161003816119889119895119896
10038161003816100381610038161003816 +10038161003816100381610038161003816119889119895119896
10038161003816100381610038161003816 le 119862212058811989521198952 (56)
This inequality and the second result of (a) yield
E (10038161003816100381610038161003816119889119895119896
minus 119889119895119896
100381610038161003816100381610038164
) le 119862221205881198952119895E (
10038161003816100381610038161003816119889119895119896minus 119889
119895119896
100381610038161003816100381610038162
) le 119862241205881198952119895 1
119899
(57)
Using 2119895 le 21198951 le 119899 the proof of (b) is completed(c) We will use the Liebscher inequality described in
Lemma 9 Let us set
119880119894= 119902 (120595
119895119896 119881
119894) minus E (119902 (120595
119895119896 119881
1)) (58)
We have E(1198801) = 0 and by(H2)-(i) and 2119895 le 21198951 le
119899(ln 119899)3
1003816100381610038161003816119880119894
1003816100381610038161003816 le 2 sup119909isin1198811(Ω)
10038161003816100381610038161003816119902 (120595119895119896
119909)10038161003816100381610038161003816 le 119862212058811989521198952 le 1198622120588119895
radic119899
(ln 119899)3
(59)
(so 119872 = 1198622120588119895radic119899(ln 119899)3)Proceeding as for the proofs of the bounds in (a) for any
integer 119897 le 119862 ln 119899 since 2minus119895 le 2minus1198950 le 2(ln 119899)minus1 we show that
V (119897
sum119894=1
119880119894)
= V (119897
sum119894=1
119902 (120595119895119896
119881119894)) le 11986222120588119895 (119897 + 11989722minus119895) le 11986222120588119895119897
(60)
Therefore
119863119898
= max119897isin12119898
V (119897
sum119894=1
119880119894) le 11986222120588119895119898 (61)
Owing to Lemma 9 applied with 1198801 119880
119899 120582 = 120581120582
1198952
119898 = [radic120581 ln 119899] 119872 = 1198622120588119895radic119899(ln 119899)3 and the bound (61) weobtain
P(10038161003816100381610038161003816119889119895119896
minus 119889119895119896
10038161003816100381610038161003816 ge120581120582
119895
2)
le 119862(exp(minus11986212058121205822
119895119899
119863119898119898 + 120581120582
119895119898119872
) +119872
120582119895
119899119890minus120579119898)
le 119862( exp (minus119862((120581222120588119895 ln 119899)
times (22120588119895 + 1205812120588119895radic (ln 119899)
119899
times [radic120581 ln 119899] 2120588119895
radic119899
(ln 119899)3)
minus1
))
+ radic119899(ln 119899)3
(ln 119899) 119899119899119890minus120579[radic120581 ln 119899])
le 119862(119899minus1198621205812(1+120581
32) + 1198992minus120579radic120581)
(62)
Taking 120581 large enough the last term is bounded by1198621198994Thiscompletes the proof of(c)
This completes the proof of Proposition 10
Proof of Theorem 3 Theorem 3 can be proved by combiningarguments of ([36] Theorem 51) and ([51] Theorem 42) Itis close to ([30] Proof of Theorem 2) by taking 120579 rarr infin Theinterested reader can find the details below
We consider the following wavelet decomposition for 119891
119891 (119909) = sum119896isinΛ 1198950
1198881198950 119896
1206011198950 119896
(119909) +infin
sum119895=1198950
sum119896isinΛ 119895
119889119895119896
120595119895119896
(119909) (63)
where 1198881198950 119896
= int1
0119891(119909)120601
1198950 119896(119909)119889119909 and 119889
119895119896= int
1
0119891(119909)120595
119895119896(119909)119889119909
Using the orthonormality of the wavelet basis B theMISE of 119891 can be expressed as
E (10038171003817100381710038171003817119891 minus 119891
100381710038171003817100381710038172
2) = 119875 + 119876 + 119877 (64)
where
119875 = sum119896isinΛ 1198950
E (100381610038161003816100381610038161198881198950 119896 minus 119888
1198950 119896
100381610038161003816100381610038162
)
119876 =
1198951
sum119895=1198950
sum119896isinΛ 119895
E(100381610038161003816100381610038161003816119889
1198951198961|119889119895119896|ge120581120582119895
minus 119889119895119896
100381610038161003816100381610038161003816
2
)
119877 =infin
sum119895=1198951+1
sum119896isinΛ 119895
1198892
119895119896
(65)
8 Journal of Probability and Statistics
Let us now investigate sharp upper bounds for 119875 119877 and119876 successively
Upper Bound for 119875 The point (a) of Proposition 10 and2119904(2119904 + 2120588 + 1) lt 1 yield
119875 le 11986221198950
119899le 119862
ln 119899
119899le 119862(
ln 119899
119899)
2119904(2119904+2120588+1)
(66)
Upper Bound for 119877(i) For 119903 ge 1 and119901 ge 2 we have119891 isin 119861119904
119901119903(119872) sube 119861119904
2infin(119872)
Using 2119904(2119904 + 2120588 + 1) lt 2119904(2120588 + 1) we obtain
119877 le 119862infin
sum119895=1198951+1
2minus2119895119904 le 119862((ln 119899)3
119899)
2119904(2120588+1)
le 119862(ln 119899
119899)
2119904(2119904+2120588+1)
(67)
(ii) For 119903 ge 1 and 119901 isin [1 2) we have 119891 isin 119861119904
119901119903(119872) sube
119861119904+12minus1119901
2infin(119872) The condition 119904 gt (2120588 + 1)119901 implies
that (119904 + 12 minus 1119901)(2120588 + 1) gt 119904(2119904 + 2120588 + 1) Thus
119877 le 119862infin
sum119895=1198951+1
2minus2119895(119904+12minus1119901)
le 119862((ln 119899)3
119899)
2(119904+12minus1119901)(2120588+1)
le 119862(ln 119899
119899)
2119904(2119904+2120588+1)
(68)
Hence for 119903 ge 1 119901 ge 2 and 119904 gt 0 or 119901 isin [1 2) and119904 gt (2120588 + 1)119901 we have
119877 le 119862(ln 119899
119899)
2119904(2119904+2120588+1)
(69)
Upper Bound for 119876 Adopting the notation119863119895119896
= 119889119895119896
minus 119889119895119896
119876 can be written as
119876 =4
sum119894=1
119876119894 (70)
where
1198761=
1198951
sum119895=1198950
sum119896isinΛ 119895
E (10038161003816100381610038161003816119863119895119896
100381610038161003816100381610038162
1|119889119895119896|ge120581120582119895 |119889119895119896|lt1205811205821198952
)
1198762=
1198951
sum119895=1198950
sum119896isinΛ 119895
E (10038161003816100381610038161003816119863119895119896
100381610038161003816100381610038162
1|119889119895119896|ge120581120582119895 |119889119895119896|ge1205811205821198952
)
1198763=
1198951
sum119895=1198950
sum119896isinΛ 119895
E (1198892
1198951198961|119889119895119896|lt120581120582119895 |119889119895119896|ge2120581120582119895
)
1198764=
1198951
sum119895=1198950
sum119896isinΛ 119895
E (1198892
1198951198961|119889119895119896|lt120581120582119895 |119889119895119896|lt2120581120582119895
)
(71)
Upper Bound for 1198761
+ 1198763 Owing to the inequalities
1|119889119895119896|lt120581120582119895 |119889119895119896|ge2120581120582119895
le 1|119895119896|gt1205811205821198952
1|119889119895119896|ge120581120582119895 |119889119895119896|lt1205811205821198952
le
1|119895119896|gt1205811205821198952
and 1|119889119895119896|lt120581120582119895 |119889119895119896|ge2120581120582119895
le 1|119889119895119896|le2|119895119896|
theCauchy-Schwarz inequality and the points(b) and (c) ofProposition 10 we have
1198761+ 119876
3le 119862
1198951
sum119895=1198950
sum119896isinΛ 119895
E (10038161003816100381610038161003816119863119895119896
100381610038161003816100381610038162
1|119895119896|gt1205811205821198952
)
le 119862
1198951
sum119895=1198950
sum119896isinΛ 119895
(E (10038161003816100381610038161003816119863119895119896
100381610038161003816100381610038164
))12
(P(10038161003816100381610038161003816119863119895119896
10038161003816100381610038161003816 gt120581120582
119895
2))
12
le 1198621
1198992
1198951
sum119895=1198950
2119895(1+2120588) le 1198621
119899le 119862(
ln 119899
119899)
2119904(2119904+2120588+1)
(72)
Upper Bound for 1198762 It follows from the point(a) of
Proposition 10 that
1198762le
1198951
sum119895=1198950
sum119896isinΛ 119895
E (10038161003816100381610038161003816119863119895119896
100381610038161003816100381610038162
) 1|119889119895119896|ge1205811205821198952
le 1198621
119899
1198951
sum119895=j0
22120588119895 sum119896isinΛ 119895
1|119889119895119896|gt1205811205821198952
(73)
Let us now introduce the integer 119895lowastdefined by
2119895lowast = [(119899
ln 119899)
1(2119904+2120588+1)
] (74)
Note that 119895lowastisin 119895
0 119895
1 for 119899 large enough
Then 1198762can be bounded as
1198762le 119876
21+ 119876
22 (75)
where
11987621
= 1198621
119899
119895lowast
sum119895=1198950
22120588119895 sum119896isinΛ 119895
1|119889119895119896|gt1205811205821198952
11987622
= 1198621
119899
1198951
sum119895=119895lowast+1
22120588119895 sum119896isinΛ 119895
1|119889119895119896|gt1205811205821198952
(76)
On the one hand we have
11987621
le 119862ln 119899
119899
119895lowast
sum119895=1198950
2119895(1+2120588) le 119862(ln 119899
119899)
2119904(2119904+2120588+1)
(77)
On the other hand we have the following
Journal of Probability and Statistics 9
(i) For 119903 ge 1 and 119901 ge 2 the Markov inequality and 119891 isin119861119904
119901119903(119872) sube 119861119904
2infin(119872) yield
11987622
le 119862ln 119899
119899
1198951
sum119895=119895lowast+1
22120588119895 1
1205822
119895
sum119896isinΛ 119895
1198892
119895119896le 119862
infin
sum119895=119895lowast+1
sum119896isinΛ 119895
1198892
119895119896
le 119862infin
sum119895=119895lowast+1
2minus2119895119904 le 119862(ln 119899
119899)
2119904(2119904+2120588+1)
(78)
(ii) For 119903 ge 1 119901 isin [1 2) and 119904 gt (2120588 + 1)119901 the Markovinequality 119891 isin 119861119904
119901119903(119872) and (2119904 + 2120588 + 1)(2 minus 119901)2 +
(119904 + 12 minus 1119901 + 120588 minus 2120588119901)119901 = 2119904 imply that
11987622
le 119862ln 119899
119899
1198951
sum119895=119895lowast+1
22120588119895 1
120582119901
119895
sum119896isinΛ 119895
10038161003816100381610038161003816119889119895119896
10038161003816100381610038161003816119901
le 119862(ln 119899
119899)
(2minus119901)2 infin
sum119895=119895lowast+1
2119895120588(2minus119901)2minus119895(119904+12minus1119901)119901
le 119862(ln 119899
119899)
(2minus119901)2
2minus119895lowast(119904+12minus1119901+120588minus2120588119901)119901
le 119862(ln 119899
119899)
2119904(2119904+2120588+1)
(79)
Therefore for 119903 ge 1 119901 ge 2 and 119904 gt 0 or 119901 isin [1 2) and119904 gt (2120588 + 1)119901 we have
1198762le C(
ln 119899
119899)
2119904(2119904+2120588+1)
(80)
Upper Bound for 1198764 We have
1198764le
1198951
sum119895=1198950
sum119896isinΛ 119895
1198892
1198951198961|119889119895119896|lt2120581120582119895
(81)
Let 119895lowastbe the integer (74) Then 119876
4can be bound as
1198764le 119876
41+ 119876
42 (82)
where
11987641
=
119895lowast
sum119895=1198950
sum119896isinΛ 119895
1198892
1198951198961|119889119895119896|lt2120581120582119895
11987642
=
1198951
sum119895=119895lowast+1
sum119896isinΛ 119895
1198892
1198951198961|119889119895119896|lt2120581120582119895
(83)
On the one hand we have
11987641
le 119862
119895lowast
sum119895=1198950
21198951205822
119895= 119862
ln 119899
119899
119895lowast
sum119895=1198950
2119895(1+2120588) le 119862(ln 119899
119899)
2119904(2119904+2120588+1)
(84)
On the other hand we have the following
(i) For 119903 ge 1 and 119901 ge 2 since 119891 isin 119861119904
119901119903(119872) sube 119861119904
2infin(119872)
we have
11987642
leinfin
sum119895=119895lowast+1
sum119896isinΛ 119895
1198892
119895119896le 119862
infin
sum119895=119895lowast+1
2minus2119895119904 le 119862(ln 119899
119899)
2119904(2119904+2120588+1)
(85)
(ii) For 119903 ge 1 119901 isin [1 2) and 119904 gt (2120588 + 1)119901 owing to theMarkov inequality 119891 isin 119861119904
119901119903(119872) and (2119904 + 2120588 + 1)(2 minus
119901)2 + (119904 + 12 minus 1119901 + 120588 minus 2120588119901)119901 = 2119904 we get
11987642
le 119862
1198951
sum119895=119895lowast+1
1205822minus119901
119895sum
119896isinΛ 119895
10038161003816100381610038161003816119889119895119896
10038161003816100381610038161003816119901
= 119862(ln 119899
119899)
(2minus119901)2 1198951
sum119895=119895lowast+1
2119895120588(2minus119901) sum119896isinΛ 119895
10038161003816100381610038161003816119889119895119896
10038161003816100381610038161003816119901
le 119862(ln 119899
119899)
(2minus119901)2 infin
sum119895=119895lowast+1
2119895120588(2minus119901)2minus119895(119904+12minus1119901)119901
le 119862(ln 119899
119899)
(2minus119901)2
2minus119895lowast(119904+12minus1119901+120588minus2120588119901)119901
le 119862(ln 119899
119899)
2119904(2119904+2120588+1)
(86)
So for 119903 ge 1 119901 ge 2 and 119904 gt 0 or 119901 isin [1 2) and 119904 gt(2120588 + 1)119901 we have
1198764le 119862(
ln 119899
119899)
2119904(2119904+2120588+1)
(87)
Putting (70) (72) (80) and (87) together for 119903 ge 1 119901 ge 2and 119904 gt 0 or 119901 isin [1 2) and 119904 gt (2120588 + 1)119901 we obtain
119876 le 119862(ln 119899
119899)
2119904(2119904+2120588+1)
(88)
Combining (64) (66) (69) and (88) we complete theproof of Theorem 3
Proof of Theorem 4 The proof of Theorem 4 is a direct appli-cation ofTheorem 3 under (G1)ndash(G5) the function 119902 definedby (24) satisfies (H1) see ([42] equation (2)) and (H2) (i) see([42] Lemma 6) (ii) see ([42] equation (11)) and (iii) see([30] Proof of Proposition 61) with 120588 = 120575
Proof of Theorem 5 The proof ofTheorem 5 is a consequenceofTheorem 3 under (J1)ndash(J4) the function 119902 defined by (30)satisfies (H1) and (H2) (i)-(ii) see ([31] Proposition 1) and(iii) see ([52] equation (26)) with 120588 = 120575
Proof of Theorem 6 Set V(119909) = 119891(119909)119892(119909) Following themethodology of [49] we have
119891 (119909) minus 119891 (119909) = 119878 (119909) minus 119879 (119909) (89)
10 Journal of Probability and Statistics
where
119878 (119909)
=1
119892 (119909)(V (119909) minus V (119909)
+119891 (119909) (119892 (119909) minus 119892 (119909))) 1|119892(119909)|ge119888lowast2
119879 (119909) = 119891 (119909) 1|119892(119909)|lt119888lowast2
(90)
Using (K3) and the indicator function we have
|119878 (119909)| le 119862 (|V (119909) minus V (119909)| + 1003816100381610038161003816119892 (119909) minus 119892 (119909)1003816100381610038161003816) (91)
It follows from |119892(119909)| lt 119888lowast2 cap |119892(119909)| gt 119888
lowast sube |119892(119909) minus
119892(119909)| gt 119888lowast2 (K3) (K4) and the Markov inequality that
|119879 (119909)| le 1198621|119892(119909)minus119892(119909)|gt119888lowast2
le 1198621003816100381610038161003816119892 (119909) minus 119892 (119909)
1003816100381610038161003816 (92)
The triangular inequality yields
10038161003816100381610038161003816119891 (119909) minus 119891 (119909)10038161003816100381610038161003816 le 119862 (|V (119909) minus V (119909)| + 1003816100381610038161003816119892 (119909) minus 119892 (119909)
1003816100381610038161003816) (93)
The elementary inequality (119886 + 119887)2 le 2(1198862 + 1198872) implies that
E (10038171003817100381710038171003817119891 minus 119891
100381710038171003817100381710038172
2) le 119862 (E (V minus V2
2) + E (
1003817100381710038171003817119892 minus 11989210038171003817100381710038172
2)) (94)
We now bound this two MISEs via Theorem 3
Upper Bound for the MISE of V Under (K1)ndash(K6) thefunction 119902 defined by (38) satisfies the following
(H1) With V instead of 119891 since 1205851and 119883
1are independent
with E(1205851) = 0
E (119902 (120574119895119896
1198811))
= E (1198841120574119895119896
(1198831)) = E (119891 (119883
1) 120574
119895119896(119883
1))
= int1
0
119891 (119909) 120574119895119896
(119909) 119892 (119909) 119889119909 = int1
0
V (119909) 120574119895119896
(119909) 119889119909
(95)
(H2) (i)-(ii)-(iii) with 120588 = 0
(i) since 1198841(Ω) is bounded thanks to (K2) and (K3) say
|1198841| le 119872 with 119872 gt 0 we have
sup(119909119909lowast)isin1198811(Ω)
10038161003816100381610038161003816119902 (120574119895119896
(119909 119909lowast))
10038161003816100381610038161003816
= sup(119909119909lowast)isin[minus119872119872]times[01]
10038161003816100381610038161003816119909120574119895119896(119909
lowast)10038161003816100381610038161003816
le 119872 sup119909lowastisin[01]
10038161003816100381610038161003816120574119895119896(119909
lowast)10038161003816100381610038161003816 le 11986221198952
(96)
(ii) using the boundedness of 1198841(Ω) then (K4) we have
E (10038161003816100381610038161003816119902(120574119895119896
1198811)100381610038161003816100381610038162
) = E (1198842
1(120574
119895119896(119883
1))
2
)
le 119862E ((120574119895119896
(1198831))
2
)
= 119862int1
0
(120574119895119896
(119909))2
119892 (119909) 119889119909
le 119862int1
0
(120574119895119896
(119909))2
119889119909 le 119862
(97)
(iii) using the boundedness of 1198841(Ω) and making the
change of variables 119910 = 2119895119909 minus 119896 we obtain
int1198811(Ω)
10038161003816100381610038161003816119902 (120574119895119896
119909)10038161003816100381610038161003816 119889119909
= (int119872
minus119872
|119909| 119889119909)(int1
0
10038161003816100381610038161003816120574119895119896(119909
lowast)10038161003816100381610038161003816 119889119909lowast
)
= 1198722 int1
0
10038161003816100381610038161003816120574119895119896(119909)
10038161003816100381610038161003816 119889119909 le 1198622minus1198952
(98)
We conclude by applying Lemma 2 with 120588 = 0 (K5) and (K6)imply (F1) and the previous inequality implies (F2)
Therefore assuming that V isin 119861119904
119901119903(119872) with 119903 ge 1 119901 ge 2
and 119904 isin (0119873) or 119901 isin [1 2) and 119904 isin (1119901119873) Theorem 3proves the existence of a constant 119862 gt 0 satisfying
E (V minus V22) le 119862(
ln 119899
119899)
2119904(2119904+1)
(99)
Upper Bound for theMISE of 119892 Under (K1)ndash(K6) proceedingas the previous point we show that the function 119902 defined by(39) satisfies (H1) with 119892 instead of 119891 and 119883
119905instead of 119881
119905
and(H2) (i)-(ii)-(iii) with 120588 = 0Therefore assuming that 119892 isin 119861119904
119901119903(119872) with 119903 ge 1 119901 ge 2
and 119904 isin (0119873) or 119901 isin [1 2) and 119904 isin (1119901119873) Theorem 3proves the existance of a constant 119862 gt 0 satisfying
E (1003817100381710038171003817119892 minus 119892
10038171003817100381710038172
2) le 119862(
ln 119899
119899)
2119904(2119904+1)
(100)
Combining (94) (99) and (100) we end the proof ofTheorem 6
Conflict of Interests
The author declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
The author is thankful to the reviewers for their commentswhich have helped in improving the presentation
Journal of Probability and Statistics 11
References
[1] P DoukhanMixing Properties and Examples vol 85 of LectureNotes in Statistics Springer New York NY USA 1994
[2] M Carrasco and X Chen ldquoMixing and moment properties ofvarious GARCH and stochastic volatility modelsrdquo EconometricTheory vol 18 no 1 pp 17ndash39 2002
[3] R C Bradley Introduction to Strong Mixing Conditions Volume1 2 3 Kendrick Press 2007
[4] P M Robinson ldquoNonparametric estimators for time seriesrdquoJournal of Time Series Analysis vol 4 pp 185ndash207 1983
[5] G G Roussas ldquoNonparametric estimation inmixing sequencesof random variablesrdquo Journal of Statistical Planning and Infer-ence vol 18 no 2 pp 135ndash149 1987
[6] G G Roussas ldquoNonparametric regression estimation undermixing conditionsrdquo Stochastic Processes and their Applicationsvol 36 no 1 pp 107ndash116 1990
[7] Y K Truong and C J Stone ldquoNonparametric function estima-tion involving time seriesrdquo The Annals of Statistics vol 20 pp77ndash97 1992
[8] L H Tran ldquoNonparametric function estimation for time seriesby local average estimatorsrdquoThe Annals of Statistics vol 21 pp1040ndash1057 1993
[9] E Masry ldquoMultivariate local polynomial regression for timeseries uniform strong consistency and ratesrdquo Journal of TimeSeries Analysis vol 17 no 6 pp 571ndash599 1996
[10] E Masry ldquoMultivariate regression estimation local polynomialfitting for time seriesrdquo Stochastic Processes and their Applica-tions vol 65 no 1 pp 81ndash101 1996
[11] E Masry and J Fan ldquoLocal polynomial estimation of regressionfunctions for mixing processesrdquo Scandinavian Journal of Statis-tics vol 24 no 2 pp 165ndash179 1997
[12] D Bosq Nonparametric Statistics for Stochastic Processes Esti-mation and Prediction vol 110 of Lecture Notes in StatisticsSpringer New York NY USA 1998
[13] E Liebscher ldquoEstimation of the density and the regressionfunction under mixing conditionsrdquo Statistics and Decisions vol19 no 1 pp 9ndash26 2001
[14] D L Donoho and I M Johnstone ldquoIdeal spatial adaptation bywavelet shrinkagerdquo Biometrika vol 81 no 3 pp 425ndash455 1994
[15] D L Donoho and I M Johnstone ldquoAdapting to unknownsmoothness via wavelet shrinkagerdquo Journal of the AmericanStatistical Association vol 90 pp 1200ndash1224 1995
[16] D Donoho I Johnstone G Kerkyacharian and D PicardldquoWavelet shrinkage asymptopiardquo Journal of the Royal StatisticalSociety B vol 57 pp 301ndash369 1995
[17] D L Donoho I M Johnstone G Kerkyacharian and DPicard ldquoDensity estimation by wavelet thresholdingrdquo Annals ofStatistics vol 24 no 2 pp 508ndash539 1996
[18] A Antoniadis ldquoWavelets in statistics a review (with discus-sion)rdquo Journal of the Italian Statistical Society vol 6 pp 97ndash1441997
[19] W Hardle G Kerkyacharian D Picard and A TsybakovWavelet Approximation and Statistical Applications vol 129 ofLectures Notes in Statistics Springer New York NY USA 1998
[20] F Leblanc ldquoWavelet linear density estimator for a discrete-timestochastic process 119871
119901-lossesrdquo Statistics and Probability Letters
vol 27 no 1 pp 71ndash84 1996[21] K Tribouley andG Viennet ldquo119871
119901adaptive density estimation in
a alpha-mixing frameworkrdquo Annales de lrsquoinstitut Henri PoincareB vol 34 no 2 pp 179ndash208 1998
[22] E Masry ldquoWavelet-based estimation of multivariate regressionfunctions in besov spacesrdquo Journal of Nonparametric Statisticsvol 12 no 2 pp 283ndash308 2000
[23] P N Patil and Y K Truong ldquoAsymptotics for wavelet basedestimates of piecewise smooth regression for stationary timeseriesrdquo Annals of the Institute of Statistical Mathematics vol 53no 1 pp 159ndash178 2001
[24] H Doosti M Afshari and H A Niroumand ldquoWavelets fornonparametric stochastic regression with mixing stochasticprocessrdquo Communications in Statistics vol 37 no 3 pp 373ndash385 2008
[25] H Doosti and H A Niroumand ldquoMultivariate stochasticregression estimation by wavelet methods for stationary timeseriesrdquo Pakistan Journal of Statistics vol 27 no 1 pp 37ndash462009
[26] H Doosti M S Islam Y P Chaubey and P Gora ldquoTwo-dimensional wavelets for nonlinear autoregressive models withan application in dynamical systemrdquo Italian Journal of Pure andApplied Mathematics no 27 pp 39ndash62 2010
[27] J Cai and H Liang ldquoNonlinear wavelet density estimation fortruncated and dependent observationsrdquo International Journal ofWavelets Multiresolution and Information Processing vol 9 no4 pp 587ndash609 2011
[28] S Niu and H Liang ldquoNonlinear wavelet estimation of condi-tional density under left-truncated and 120572-mixing assumptionsrdquoInternational Journal of Wavelets Multiresolution and Informa-tion Processing vol 9 no 6 pp 989ndash1023 2011
[29] F Benatia and D Yahia ldquoNonlinear wavelet regression functionestimator for censored dependent datardquo Journal Afrika Statis-tika vol 7 no 1 pp 391ndash411 2012
[30] C Chesneau ldquoOn the adaptive wavelet deconvolution of adensity for strong mixing sequencesrdquo Journal of the KoreanStatistical Society vol 41 no 4 pp 423ndash436 2012
[31] C Chesneau ldquoWavelet estimation of a density in a GARCH-type modelrdquo Communications in Statistics vol 42 no 1 pp 98ndash117 2013
[32] C Chesneau ldquoOn the adaptive wavelet estimation of a mul-tidimensional regression function under alpha-mixing depen-dence beyond the standard assumptions on the noiserdquo Com-mentationes Mathematicae Universitatis Carolinae vol 4 pp527ndash556 2013
[33] Y P Chaubey and E Shirazi ldquoOn MISE of a nonlinear waveletestimator of the regression function based on biased data understrong mixing rdquo Communications in Statistics In press
[34] M Abbaszadeh andM Emadi ldquoWavelet density estimation andstatistical evidences role for a garch model in the weighteddistributionrdquo Applied Mathematics vol 4 no 2 pp 10ndash4162013
[35] B Delyon and A Juditsky ldquoOn minimax wavelet estimatorsrdquoApplied and Computational Harmonic Analysis vol 3 no 3 pp215ndash228 1996
[36] G Kerkyacharian and D Picard ldquoThresholding algorithmsmaxisets and well-concentrated basesrdquo Test vol 9 no 2 pp283ndash344 2000
[37] I Daubechies Ten Lectures on Wavelets SIAM 1992[38] A Cohen I Daubechies and P Vial ldquoWavelets on the interval
and fast wavelet transformsrdquo Applied and Computational Har-monic Analysis vol 1 no 1 pp 54ndash81 1993
[39] S Mallat A Wavelet Tour of Signal Processing The sparse waywith contributions fromGabriel Peyralphae ElsevierAcademicPress Amsterdam The Netherlands 3rd edition 2009
12 Journal of Probability and Statistics
[40] Y Meyer Ondelettes et Opalphaerateurs Hermann ParisFrance 1990
[41] V Genon-Catalot T Jeantheau and C Laredo ldquoStochasticvolatility models as hidden Markov models and statisticalapplicationsrdquo Bernoulli vol 6 no 6 pp 1051ndash1079 2000
[42] J Fan and J Koo ldquoWavelet deconvolutionrdquo IEEE Transactionson Information Theory vol 48 no 3 pp 734ndash747 2002
[43] A B Tsybakov Introduction alphaa lrsquoestimation non-paramalphaetrique Springer New York NY USA 2004
[44] E Masry ldquoStrong consistency and rates for deconvolutionof multivariate densities of stationary processesrdquo StochasticProcesses and their Applications vol 47 no 1 pp 53ndash74 1993
[45] R Kulik ldquoNonparametric deconvolution problem for depen-dent sequencesrdquo Electronic Journal of Statistics vol 2 pp 722ndash740 2008
[46] F Comte J Dedecker and M L Taupin ldquoAdaptive densitydeconvolution with dependent inputsrdquo Mathematical Methodsof Statistics vol 17 no 2 pp 87ndash112 2008
[47] H van Zanten and P Zareba ldquoA note on wavelet densitydeconvolution for weakly dependent datardquo Statistical Inferencefor Stochastic Processes vol 11 no 2 pp 207ndash219 2008
[48] W Hardle Applied Nonparametric Regression Cambridge Uni-versity Press Cambridge UK 1990
[49] V A Vasiliev ldquoOne investigation method of a ratios typeestimatorsrdquo in Proceedings of the 16th IFAC Symposium onSystem Identialphacation pp 1ndash6 Brussels Belgium July 2012(in progress)
[50] Y Davydov ldquoThe invariance principle for stationary processesrdquoTheory of Probability and Its Applications vol 15 no 3 pp 498ndash509 1970
[51] C Chesneau ldquoWavelet estimation via block thresholding aminimax study under 119871119901-riskrdquo Statistica Sinica vol 18 no 3pp 1007ndash1024 2008
[52] C Chesneau and H Doosti ldquoWavelet linear density estimationfor a GARCH model under various dependence structuresrdquoJournal of the Iranian Statistical Society vol 11 no 1 pp 1ndash212012
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
4 Journal of Probability and Statistics
(G1) The support of 119891 is [0 1](G2) There exists a constant 119862 gt 0 such that
sup119909isinR
119891 (119909) le 119862 lt infin (18)
(G3) Let 119906 be the density of1198811We suppose that there exists
a constant 119862 gt 0 such that
sup119909isinR
119906 (119909) le 119862 (19)
(G4) For any 119898 isin Z let 119906(1198811 119881119898+1)
be the density of(119881
1 119881
119898+1) We suppose that there exists a constant
119862 gt 0 such that
sup119898isinZ
sup(119909119910)isinR2
119906(1198811 119881119898+1)
(119909 119910) le 119862 (20)
(G5) For any integrable function 120574 we define its Fouriertransform by
F (120574) (119909) = intinfin
minusinfin
120574 (119910) 119890minus119894119909119910119889119910 119909 isin R (21)
We suppose that there exist three known constants119862 gt 0119888 gt 0 and 120575 gt 1 such that for any 119909 isin R
(i) the Fourier transform of 119892 satisfies
1003816100381610038161003816F (119892) (119909)1003816100381610038161003816 ge
119888
(1 + 1199092)1205752
(22)
(ii) for any ℓ isin 0 1 2 the ℓth derivative of the Fouriertransform of 119892 satisfies
100381610038161003816100381610038161003816(F (119892) (119909))
(ℓ)100381610038161003816100381610038161003816le
119862
(1 + |119909|)120575+ℓ (23)
We are now in the position to present the result
Theorem 4 We consider the model (17) Suppose that (G1)ndash(G5) are satisfied Let 119891 be defined as in (7) with
119902 (120574 119909) =1
2120587int
infin
minusinfin
F (120574) (119910)
F (119892) (119910)119890minus119894119910119909119889119910 (24)
whereF(120574)(119910) denotes the complex conjugate ofF(120574)(119910) and120588 = 120575 (appearing in(G5))
Suppose that 119891 isin 119861119904
119901119903(119872) with 119903 ge 1 119901 ge 2 and 119904 isin
(0119873) or 119901 isin [1 2) and 119904 isin ((2120575 + 1)119901119873) Then thereexists a constant 119862 gt 0 such that
E (10038171003817100381710038171003817119891 minus 119891
100381710038171003817100381710038172
2) le 119862(
ln 119899
119899)
2119904(2119904+2120575+1)
(25)
Theorem 4 improves ([30] Proposition 51) in terms ofrate of convergence we gain a logarithmic term
Moreover it is established that in the iid settingldquo((ln 119899)119899)2119904(2119904+2120575+1)rdquo is
(i) exactly the rate of convergence attained by the hardthresholding wavelet estimator
(ii) the near optimal rate of convergence in the minimaxsense
The details can be found in Fan and Koo [42] ThusTheorem 4 can be viewed as an extension of this existingresult to the weak dependent case
42 GARCH-TypeModel We consider the strictly stationarystochastic process (119881
119905)119905isinZ where for any 119905 isin Z
119881119905= 1205902
119905119885
119905 (26)
(1205902
119905)119905isinZ is a strictly stationary stochastic process with
unknown density 119891 and (119885119905)119905isinZ is a strictly stationary
stochastic process with known density 119892 It is supposed that1205902
119905and 119885
119905are independent for any 119905 isin Z and (119881
119905)119905isinZ is a 120572-
mixing process with exponential decay rate (see Section 31for a precise definition) Our aim is to estimate 119891 via119881
1 119881
119899from (119881
119905)119905isinZ Some related works are Comte et al
[46] and Chesneau [31]We formulate the following assumptions(J1) There exists a positive integer 120575 such that
119892 (119909) =1
(120575 minus 1)(minus ln119909)
120575minus1 119909 isin [0 1] (27)
Let us remark that 119892 is the density of prod120575
119894=1119880
119894 where
1198801 119880
120575are 120575 iid random variables having the
common distributionU([0 1])(J2) The support of 119891 is [0 1] and 119891 isin L2([0 1])(J3) Let 119906 be the density of119881
1We suppose that there exists
a constant 119862 gt 0 such thatsup119909isinR
119906 (119909) le 119862 (28)
(J4) For any 119898 isin Z let 119906(1198811 119881119898+1)
be the density of(119881
1 119881
119898+1) We suppose that there exists a constant
119862 gt 0 such thatsup119898isinZ
sup(119909119910)isinR2
119906(1198811 119881119898+1)
(119909 119910) le 119862 (29)
We are now in the position to present the result
Theorem 5 We consider model (26) Suppose that (J1)ndash(J4)are satisfied Let 119891 be defined as in (7) with
119902 (120574 119909) = 119879120575(120574) (119909) (30)
where for any positive integer ℓ 119879(120574)(119909) = (119909120574(119909))1015840 and119879
ℓ(120574)(119909) = 119879(119879
ℓminus1(120574))(119909) and 120588 = 120575 (appearing in (J1))
Suppose that 119891 isin 119861119904
119901119903(119872) with 119903 ge 1 119901 ge 2 and 119904 isin
(0119873) or 119901 isin [1 2) and 119904 isin ((2120575 + 1)119901119873) Then thereexists a constant 119862 gt 0 such that
E (10038171003817100381710038171003817119891 minus 119891
100381710038171003817100381710038172
2) le 119862(
ln 119899
119899)
2119904(2119904+2120575+1)
(31)
Theorem 5 significantly improves ([31] Theorem 2) interms of rate of convergence we gain an exponent 12
Journal of Probability and Statistics 5
43 Nonparametric Regression Model We consider thestrictly stationary stochastic process (119881
119905)119905isinZ where for any
119905 isin Z 119881119905= (119884
119905 119883
119905)
119884119905= 119891 (119883
119905) + 120585
119905 (32)
(119883119905)119905isinZ is a strictly stationary stochastic process with
unknown density 119892 (120585119905)119905isinZ is a strictly stationary centered
stochastic process and119891 is the unknown regression functionIt is supposed that 119883
119905and 120585
119905are independent for any 119905 isin Z
and (119881119905)119905isinZ is a 120572-mixing process with exponential decay
rate (see Section 31 for a precise definition) Our aim is toestimate 119891 via 119881
1 119881
119899from (119881
119905)119905isinZ Applications of this
problem can be found in Hardle [48] Wavelet methods canbe found in Patil and Truong [23] Doosti et al [24] Doostiet al [26] and Doosti and Niroumand [25]
We formulate the following assumptions
(K1) The support of 119891 and 119892 is [0 1] and 119891 and 119892 isin
L2([0 1])(K2) 120585
1(Ω) is bounded
(K3) There exists a constant 119862 gt 0 such that
sup119909isin[01]
1003816100381610038161003816119891 (119909)1003816100381610038161003816 le 119862 (33)
(K4) There exist two constants 119888lowastgt 0 and 119862 gt 0 such that
119888lowastle inf
119909isin[01]
119892 (119909) sup119909isin[01]
119892 (119909) le 119862 (34)
(K5) Let 119906 be the density of1198811We suppose that there exists
a constant 119862 gt 0 such that
sup119909isinRtimes[01]
119906 (119909) le 119862 (35)
(K6) For any 119898 isin Z let 119906(1198811 119881119898+1)
be the density of(119881
1 119881
119898+1) We suppose that there exists a constant
119862 gt 0 such that
sup119898isinZ
sup(119909119910)isin(Rtimes[01])times(Rtimes[01])
119906(1198811 119881119898+1)
(119909 119910) le 119862 (36)
We are now in the position to present the result
Theorem 6 We consider the model (32) Suppose that (K1)ndash(K6) are satisfied Let 119891 be the truncated ratio estimatorConsider
119891 (119909) =V (119909)119892 (119909)
1|119892(119909)|ge119888lowast2
(37)
where
(i) V is defined as in (7) with
119902 (120574 (119909 119909lowast)) = 119909120574 (119909
lowast) (38)
and 120588 = 0
(ii) 119892 is defined as in (7) with 119883119905instead of 119881
119905
119902 (120574 119909) = 120574 (119909) (39)
and 120588 = 0(iii) 119888
lowastis the constant defined in (K4)
Suppose that 119891119892 isin 119861119904
119901119903(119872) and 119892 isin 119861119904
119901119903(119872) with 119903 ge 1
119901 ge 2 and 119904 isin (0119873) or 119901 isin [1 2) and 119904 isin (1119901119873) Thenthere exists a constant 119862 gt 0 such that
E (10038171003817100381710038171003817119891 minus 119891
100381710038171003817100381710038172
2) le 119862(
ln 119899
119899)
2119904(2119904+1)
(40)
The estimator (37) is derived by combining the procedureof Patil and Truong [23] with the truncated approach ofVasiliev [49]
Theorem 6 completes Patil and Truong [23] in terms ofrates of convergence under the MISE over Besov balls
Remark 7 The assumption(K2) can be relaxed with anotherstrategy to the one developed in Theorem 6 Some technicalelements are given in Chesneau [32]
Conclusion Considering the weak dependent case on theobservations we prove a general result on the rate of conver-gence attains by a hard wavelet thresholding estimator underthe MISE over Besov balls This result is flexible it can beapplied for a wide class of statistical models Moreover theobtained rate of convergence is sharp it can correspond to thenear optimal one in the minimax sense for the standard iidcase Some recent results on sophisticated statistical problemsare improved Thanks to its flexibility the perspectives ofapplications of our theoretical result in other contexts arenumerous
5 Proofs
In this section 119862 denotes any constant that does not dependon 119895 119896 and 119899 Its value may change from one term to anotherand may depend on 120601 or 120595
51 Key Lemmas Let us present two lemmas which will beused in the proofs
Lemma 8 shows a sharp covariance inequality under the120572-mixing condition
Lemma 8 (see [50]) Let (119882119905)119905isinZ be a strictly stationary 120572-
mixing process withmixing coefficient 120572119898119898 ge 0 and let ℎ and
119896 be two measurable functions Let 119901 gt 0 and 119902 gt 0 satisfying1119901 + 1119902 lt 1 such that E(|ℎ(119882
1)|119901) and E(|119896(119882
1)|119902) exist
Then there exists a constant 119862 gt 0 such that1003816100381610038161003816C119900V (ℎ (119882
1) 119896 (119882
119898+1))1003816100381610038161003816
le 1198621205721minus1119901minus1119902
119898(E (
1003816100381610038161003816ℎ (1198821)1003816100381610038161003816119901
))1119901
(E(1003816100381610038161003816119896 (119882
1)1003816100381610038161003816119902
))1119902
(41)
Lemma 9 below presents a concentration inequality for120572-mixing processes
6 Journal of Probability and Statistics
Lemma9 (see [13]) Let (119882119905)119905isinZ be a strictly stationary process
with the 119898th strongly mixing coefficient 120572119898 119898 ge 0 let 119899 be a
positive integer let ℎ R rarr C be a measurable function andfor any 119905 isin Z 119880
119905= ℎ(119882
119905) We assume that E(119880
1) = 0 and
there exists a constant 119872 gt 0 satisfying |1198801| le 119872 Then for
any 119898 isin 1 [1198992] and 120582 gt 0 we have
P(1
119899
100381610038161003816100381610038161003816100381610038161003816
119899
sum119894=1
119880119894
100381610038161003816100381610038161003816100381610038161003816ge 120582)
le 4 exp(minus1205822119899
16 (119863119898119898 + 1205821198721198983)
) + 32119872
120582119899120572
119898
(42)
where
119863119898
= max119897isin12119898
V (119897
sum119894=1
119880119894) (43)
52 Intermediary Results
Proof of Lemma 2 Using a standard expression of the covari-ance and (F1) as well as(F2) we obtain
10038161003816100381610038161003816C119900V (119902 (120574119895119896
119881119898+1
) 119902 (120574119895119896
1198811))
10038161003816100381610038161003816
=100381610038161003816100381610038161003816100381610038161003816int1198811(Ω)
int1198811(Ω)
119902 (120574119895119896
119909) 119902 (120574119895119896
119910)
times (119906(1198811 119881119898+1)
(119909 119910) minus 119906 (119909) 119906 (119910)) 119889119909119889119910100381610038161003816100381610038161003816100381610038161003816
le int1198811(Ω)
int1198811(Ω)
10038161003816100381610038161003816119902 (120574119895119896
119909)1003816100381610038161003816100381610038161003816100381610038161003816119902 (120574
119895119896 119910)
10038161003816100381610038161003816
times10038161003816100381610038161003816119906(1198811 119881119898+1)
(119909 119910) minus 119906 (119909) 119906 (119910)10038161003816100381610038161003816 119889119909 119889119910
le 119862(int1198811(Ω)
10038161003816100381610038161003816119902 (120574119895119896
119909)10038161003816100381610038161003816 119889119909)
2
le 119862221205881198952minus119895
(44)
This ends the proof of Lemma 2
Proposition 10 proves probability and moments inequal-ities satisfied by the estimators (8)
Proposition 10 Let 119895119896
and 120573119895119896
be defined as in (8) under(H1) and (H2) let 119895
0be (9) and let 119895
1be (10)
(a) There exists a constant 119862 gt 0 such that for any 119895 isin119895
0 119895
1 and 119896 isin Λ
119895
E (10038161003816100381610038161003816119888119895119896 minus 119888
119895119896
100381610038161003816100381610038162
) le 11986222120588119895 1
119899 (45)
E (10038161003816100381610038161003816119889119895119896
minus 119889119895119896
100381610038161003816100381610038162
) le 11986222120588119895 1
119899 (46)
(b) There exists a constant 119862 gt 0 such that for any 119895 isin119895
0 119895
1 and 119896 isin Λ
119895
E (10038161003816100381610038161003816119889119895119896
minus 119889119895119896
100381610038161003816100381610038164
) le 11986224120588119895 (47)
(c) Let 120582119895be defined as in (11) There exists a constant 119862 gt
0 such that for any 120581 large enough 119895 isin 1198950 119895
1 and
119896 isin Λ119895 we have
P(10038161003816100381610038161003816119889119895119896
minus 119889119895119896
10038161003816100381610038161003816 ge120581120582
119895
2) le 119862
1
1198994 (48)
Proof of Proposition 10 (a) Using (H1) and the stationarity of(119881
119905)119905isinZ we obtain
E (10038161003816100381610038161003816119888119895119896 minus 119888
119895119896
100381610038161003816100381610038162
) = V (119888119895119896
)
=1
1198992
119899
sum119894=1
V (119902 (120601119895119896
119881119894)) +
2
1198992
times119899
sumV=2
Vminus1
sumℓ=1
Re (C119900V (119902 (120601
119895119896 119881V) 119902 (120601
119895119896 119881
ℓ)))
=1
1198992
119899
sum119894=1
V (119902 (120601119895k 119881119894
)) +2
1198992
times119899minus1
sum119898=1
(119899 minus 119898)Re (C119900V (119902 (120601
119895119896 119881
119898+1) 119902 (120601
119895119896 119881
1)))
le1
119899(E (
10038161003816100381610038161003816119902 (120601119895119896
1198811)100381610038161003816100381610038162
)
+2119899minus1
sum119898=1
10038161003816100381610038161003816C119900V (119902 (120601119895119896
119881119898+1
) 119902 (120601119895119896
1198811))
10038161003816100381610038161003816)
(49)
By (H2)-(ii) we get
E (10038161003816100381610038161003816119902(120601119895119896
1198811)100381610038161003816100381610038162
) le 11986222120588119895 (50)
For the covariance term note that
119899minus1
sum119898=1
10038161003816100381610038161003816C119900V (119902 (120601119895119896
119881119898+1
) 119902 (120601119895119896
1198811))
10038161003816100381610038161003816 = 119860 + 119861 (51)
where
119860 =[ln 119899120579]minus1
sum119898=1
10038161003816100381610038161003816C119900V (119902 (120601119895119896
119881119898+1
) 119902 (120601119895119896
1198811))
10038161003816100381610038161003816
119861 =119899minus1
sum119898=[(ln 119899)120579]
10038161003816100381610038161003816C119900V (119902 (120601119895119896
119881119898+1
) 119902 (120601119895119896
1198811))
10038161003816100381610038161003816
(52)
It follows from (H2)-(iii) and 2minus119895 le 2minus1198950 lt 2(ln 119899)minus1 that
119860 le 119862221205881198952minus119895 [ln 119899
120579] le 11986222120588119895 (53)
Journal of Probability and Statistics 7
The Davydov inequality described in Lemma 8 with 119901 = 119902 =
4 (H2)-(i)-(ii) and 2119895 le 21198951 le 119899 give
119861 le 119862radicE (10038161003816100381610038161003816119902(120601119895119896
1198811)100381610038161003816100381610038164
)119899minus1
sum119898=[(ln 119899)120579]
radic120572119898
le 119862212058811989521198952radicE (10038161003816100381610038161003816119902(120601119895119896
1198811)100381610038161003816100381610038162
)infin
sum119898=[(ln 119899)120579]
119890minus1205791198982
= 11986222120588119895radic119899119890minus(ln 119899)2 le 11986222120588119895
(54)
Thus119899minus1
sum119898=1
10038161003816100381610038161003816C119900V (119902 (120601119895119896
119881119898+1
) 119902 (120601119895119896
1198811))
10038161003816100381610038161003816 le 11986222120588119895 (55)
Putting (49) (50) and (55) together the first point in (a)is proved The proof of the second point is identical with 120595instead of 120601
(b) Thanks to (H2)-(i) we have |119889119895119896
| le
sup119909isin1198811(Ω)
|119902(120595119895119896
119909)| le 119862212058811989521198952 It follows from thetriangular inequality and |119889
119895119896| le 119891
2le 119862 that
10038161003816100381610038161003816119889119895119896minus 119889
119895119896
10038161003816100381610038161003816 le10038161003816100381610038161003816119889119895119896
10038161003816100381610038161003816 +10038161003816100381610038161003816119889119895119896
10038161003816100381610038161003816 le 119862212058811989521198952 (56)
This inequality and the second result of (a) yield
E (10038161003816100381610038161003816119889119895119896
minus 119889119895119896
100381610038161003816100381610038164
) le 119862221205881198952119895E (
10038161003816100381610038161003816119889119895119896minus 119889
119895119896
100381610038161003816100381610038162
) le 119862241205881198952119895 1
119899
(57)
Using 2119895 le 21198951 le 119899 the proof of (b) is completed(c) We will use the Liebscher inequality described in
Lemma 9 Let us set
119880119894= 119902 (120595
119895119896 119881
119894) minus E (119902 (120595
119895119896 119881
1)) (58)
We have E(1198801) = 0 and by(H2)-(i) and 2119895 le 21198951 le
119899(ln 119899)3
1003816100381610038161003816119880119894
1003816100381610038161003816 le 2 sup119909isin1198811(Ω)
10038161003816100381610038161003816119902 (120595119895119896
119909)10038161003816100381610038161003816 le 119862212058811989521198952 le 1198622120588119895
radic119899
(ln 119899)3
(59)
(so 119872 = 1198622120588119895radic119899(ln 119899)3)Proceeding as for the proofs of the bounds in (a) for any
integer 119897 le 119862 ln 119899 since 2minus119895 le 2minus1198950 le 2(ln 119899)minus1 we show that
V (119897
sum119894=1
119880119894)
= V (119897
sum119894=1
119902 (120595119895119896
119881119894)) le 11986222120588119895 (119897 + 11989722minus119895) le 11986222120588119895119897
(60)
Therefore
119863119898
= max119897isin12119898
V (119897
sum119894=1
119880119894) le 11986222120588119895119898 (61)
Owing to Lemma 9 applied with 1198801 119880
119899 120582 = 120581120582
1198952
119898 = [radic120581 ln 119899] 119872 = 1198622120588119895radic119899(ln 119899)3 and the bound (61) weobtain
P(10038161003816100381610038161003816119889119895119896
minus 119889119895119896
10038161003816100381610038161003816 ge120581120582
119895
2)
le 119862(exp(minus11986212058121205822
119895119899
119863119898119898 + 120581120582
119895119898119872
) +119872
120582119895
119899119890minus120579119898)
le 119862( exp (minus119862((120581222120588119895 ln 119899)
times (22120588119895 + 1205812120588119895radic (ln 119899)
119899
times [radic120581 ln 119899] 2120588119895
radic119899
(ln 119899)3)
minus1
))
+ radic119899(ln 119899)3
(ln 119899) 119899119899119890minus120579[radic120581 ln 119899])
le 119862(119899minus1198621205812(1+120581
32) + 1198992minus120579radic120581)
(62)
Taking 120581 large enough the last term is bounded by1198621198994Thiscompletes the proof of(c)
This completes the proof of Proposition 10
Proof of Theorem 3 Theorem 3 can be proved by combiningarguments of ([36] Theorem 51) and ([51] Theorem 42) Itis close to ([30] Proof of Theorem 2) by taking 120579 rarr infin Theinterested reader can find the details below
We consider the following wavelet decomposition for 119891
119891 (119909) = sum119896isinΛ 1198950
1198881198950 119896
1206011198950 119896
(119909) +infin
sum119895=1198950
sum119896isinΛ 119895
119889119895119896
120595119895119896
(119909) (63)
where 1198881198950 119896
= int1
0119891(119909)120601
1198950 119896(119909)119889119909 and 119889
119895119896= int
1
0119891(119909)120595
119895119896(119909)119889119909
Using the orthonormality of the wavelet basis B theMISE of 119891 can be expressed as
E (10038171003817100381710038171003817119891 minus 119891
100381710038171003817100381710038172
2) = 119875 + 119876 + 119877 (64)
where
119875 = sum119896isinΛ 1198950
E (100381610038161003816100381610038161198881198950 119896 minus 119888
1198950 119896
100381610038161003816100381610038162
)
119876 =
1198951
sum119895=1198950
sum119896isinΛ 119895
E(100381610038161003816100381610038161003816119889
1198951198961|119889119895119896|ge120581120582119895
minus 119889119895119896
100381610038161003816100381610038161003816
2
)
119877 =infin
sum119895=1198951+1
sum119896isinΛ 119895
1198892
119895119896
(65)
8 Journal of Probability and Statistics
Let us now investigate sharp upper bounds for 119875 119877 and119876 successively
Upper Bound for 119875 The point (a) of Proposition 10 and2119904(2119904 + 2120588 + 1) lt 1 yield
119875 le 11986221198950
119899le 119862
ln 119899
119899le 119862(
ln 119899
119899)
2119904(2119904+2120588+1)
(66)
Upper Bound for 119877(i) For 119903 ge 1 and119901 ge 2 we have119891 isin 119861119904
119901119903(119872) sube 119861119904
2infin(119872)
Using 2119904(2119904 + 2120588 + 1) lt 2119904(2120588 + 1) we obtain
119877 le 119862infin
sum119895=1198951+1
2minus2119895119904 le 119862((ln 119899)3
119899)
2119904(2120588+1)
le 119862(ln 119899
119899)
2119904(2119904+2120588+1)
(67)
(ii) For 119903 ge 1 and 119901 isin [1 2) we have 119891 isin 119861119904
119901119903(119872) sube
119861119904+12minus1119901
2infin(119872) The condition 119904 gt (2120588 + 1)119901 implies
that (119904 + 12 minus 1119901)(2120588 + 1) gt 119904(2119904 + 2120588 + 1) Thus
119877 le 119862infin
sum119895=1198951+1
2minus2119895(119904+12minus1119901)
le 119862((ln 119899)3
119899)
2(119904+12minus1119901)(2120588+1)
le 119862(ln 119899
119899)
2119904(2119904+2120588+1)
(68)
Hence for 119903 ge 1 119901 ge 2 and 119904 gt 0 or 119901 isin [1 2) and119904 gt (2120588 + 1)119901 we have
119877 le 119862(ln 119899
119899)
2119904(2119904+2120588+1)
(69)
Upper Bound for 119876 Adopting the notation119863119895119896
= 119889119895119896
minus 119889119895119896
119876 can be written as
119876 =4
sum119894=1
119876119894 (70)
where
1198761=
1198951
sum119895=1198950
sum119896isinΛ 119895
E (10038161003816100381610038161003816119863119895119896
100381610038161003816100381610038162
1|119889119895119896|ge120581120582119895 |119889119895119896|lt1205811205821198952
)
1198762=
1198951
sum119895=1198950
sum119896isinΛ 119895
E (10038161003816100381610038161003816119863119895119896
100381610038161003816100381610038162
1|119889119895119896|ge120581120582119895 |119889119895119896|ge1205811205821198952
)
1198763=
1198951
sum119895=1198950
sum119896isinΛ 119895
E (1198892
1198951198961|119889119895119896|lt120581120582119895 |119889119895119896|ge2120581120582119895
)
1198764=
1198951
sum119895=1198950
sum119896isinΛ 119895
E (1198892
1198951198961|119889119895119896|lt120581120582119895 |119889119895119896|lt2120581120582119895
)
(71)
Upper Bound for 1198761
+ 1198763 Owing to the inequalities
1|119889119895119896|lt120581120582119895 |119889119895119896|ge2120581120582119895
le 1|119895119896|gt1205811205821198952
1|119889119895119896|ge120581120582119895 |119889119895119896|lt1205811205821198952
le
1|119895119896|gt1205811205821198952
and 1|119889119895119896|lt120581120582119895 |119889119895119896|ge2120581120582119895
le 1|119889119895119896|le2|119895119896|
theCauchy-Schwarz inequality and the points(b) and (c) ofProposition 10 we have
1198761+ 119876
3le 119862
1198951
sum119895=1198950
sum119896isinΛ 119895
E (10038161003816100381610038161003816119863119895119896
100381610038161003816100381610038162
1|119895119896|gt1205811205821198952
)
le 119862
1198951
sum119895=1198950
sum119896isinΛ 119895
(E (10038161003816100381610038161003816119863119895119896
100381610038161003816100381610038164
))12
(P(10038161003816100381610038161003816119863119895119896
10038161003816100381610038161003816 gt120581120582
119895
2))
12
le 1198621
1198992
1198951
sum119895=1198950
2119895(1+2120588) le 1198621
119899le 119862(
ln 119899
119899)
2119904(2119904+2120588+1)
(72)
Upper Bound for 1198762 It follows from the point(a) of
Proposition 10 that
1198762le
1198951
sum119895=1198950
sum119896isinΛ 119895
E (10038161003816100381610038161003816119863119895119896
100381610038161003816100381610038162
) 1|119889119895119896|ge1205811205821198952
le 1198621
119899
1198951
sum119895=j0
22120588119895 sum119896isinΛ 119895
1|119889119895119896|gt1205811205821198952
(73)
Let us now introduce the integer 119895lowastdefined by
2119895lowast = [(119899
ln 119899)
1(2119904+2120588+1)
] (74)
Note that 119895lowastisin 119895
0 119895
1 for 119899 large enough
Then 1198762can be bounded as
1198762le 119876
21+ 119876
22 (75)
where
11987621
= 1198621
119899
119895lowast
sum119895=1198950
22120588119895 sum119896isinΛ 119895
1|119889119895119896|gt1205811205821198952
11987622
= 1198621
119899
1198951
sum119895=119895lowast+1
22120588119895 sum119896isinΛ 119895
1|119889119895119896|gt1205811205821198952
(76)
On the one hand we have
11987621
le 119862ln 119899
119899
119895lowast
sum119895=1198950
2119895(1+2120588) le 119862(ln 119899
119899)
2119904(2119904+2120588+1)
(77)
On the other hand we have the following
Journal of Probability and Statistics 9
(i) For 119903 ge 1 and 119901 ge 2 the Markov inequality and 119891 isin119861119904
119901119903(119872) sube 119861119904
2infin(119872) yield
11987622
le 119862ln 119899
119899
1198951
sum119895=119895lowast+1
22120588119895 1
1205822
119895
sum119896isinΛ 119895
1198892
119895119896le 119862
infin
sum119895=119895lowast+1
sum119896isinΛ 119895
1198892
119895119896
le 119862infin
sum119895=119895lowast+1
2minus2119895119904 le 119862(ln 119899
119899)
2119904(2119904+2120588+1)
(78)
(ii) For 119903 ge 1 119901 isin [1 2) and 119904 gt (2120588 + 1)119901 the Markovinequality 119891 isin 119861119904
119901119903(119872) and (2119904 + 2120588 + 1)(2 minus 119901)2 +
(119904 + 12 minus 1119901 + 120588 minus 2120588119901)119901 = 2119904 imply that
11987622
le 119862ln 119899
119899
1198951
sum119895=119895lowast+1
22120588119895 1
120582119901
119895
sum119896isinΛ 119895
10038161003816100381610038161003816119889119895119896
10038161003816100381610038161003816119901
le 119862(ln 119899
119899)
(2minus119901)2 infin
sum119895=119895lowast+1
2119895120588(2minus119901)2minus119895(119904+12minus1119901)119901
le 119862(ln 119899
119899)
(2minus119901)2
2minus119895lowast(119904+12minus1119901+120588minus2120588119901)119901
le 119862(ln 119899
119899)
2119904(2119904+2120588+1)
(79)
Therefore for 119903 ge 1 119901 ge 2 and 119904 gt 0 or 119901 isin [1 2) and119904 gt (2120588 + 1)119901 we have
1198762le C(
ln 119899
119899)
2119904(2119904+2120588+1)
(80)
Upper Bound for 1198764 We have
1198764le
1198951
sum119895=1198950
sum119896isinΛ 119895
1198892
1198951198961|119889119895119896|lt2120581120582119895
(81)
Let 119895lowastbe the integer (74) Then 119876
4can be bound as
1198764le 119876
41+ 119876
42 (82)
where
11987641
=
119895lowast
sum119895=1198950
sum119896isinΛ 119895
1198892
1198951198961|119889119895119896|lt2120581120582119895
11987642
=
1198951
sum119895=119895lowast+1
sum119896isinΛ 119895
1198892
1198951198961|119889119895119896|lt2120581120582119895
(83)
On the one hand we have
11987641
le 119862
119895lowast
sum119895=1198950
21198951205822
119895= 119862
ln 119899
119899
119895lowast
sum119895=1198950
2119895(1+2120588) le 119862(ln 119899
119899)
2119904(2119904+2120588+1)
(84)
On the other hand we have the following
(i) For 119903 ge 1 and 119901 ge 2 since 119891 isin 119861119904
119901119903(119872) sube 119861119904
2infin(119872)
we have
11987642
leinfin
sum119895=119895lowast+1
sum119896isinΛ 119895
1198892
119895119896le 119862
infin
sum119895=119895lowast+1
2minus2119895119904 le 119862(ln 119899
119899)
2119904(2119904+2120588+1)
(85)
(ii) For 119903 ge 1 119901 isin [1 2) and 119904 gt (2120588 + 1)119901 owing to theMarkov inequality 119891 isin 119861119904
119901119903(119872) and (2119904 + 2120588 + 1)(2 minus
119901)2 + (119904 + 12 minus 1119901 + 120588 minus 2120588119901)119901 = 2119904 we get
11987642
le 119862
1198951
sum119895=119895lowast+1
1205822minus119901
119895sum
119896isinΛ 119895
10038161003816100381610038161003816119889119895119896
10038161003816100381610038161003816119901
= 119862(ln 119899
119899)
(2minus119901)2 1198951
sum119895=119895lowast+1
2119895120588(2minus119901) sum119896isinΛ 119895
10038161003816100381610038161003816119889119895119896
10038161003816100381610038161003816119901
le 119862(ln 119899
119899)
(2minus119901)2 infin
sum119895=119895lowast+1
2119895120588(2minus119901)2minus119895(119904+12minus1119901)119901
le 119862(ln 119899
119899)
(2minus119901)2
2minus119895lowast(119904+12minus1119901+120588minus2120588119901)119901
le 119862(ln 119899
119899)
2119904(2119904+2120588+1)
(86)
So for 119903 ge 1 119901 ge 2 and 119904 gt 0 or 119901 isin [1 2) and 119904 gt(2120588 + 1)119901 we have
1198764le 119862(
ln 119899
119899)
2119904(2119904+2120588+1)
(87)
Putting (70) (72) (80) and (87) together for 119903 ge 1 119901 ge 2and 119904 gt 0 or 119901 isin [1 2) and 119904 gt (2120588 + 1)119901 we obtain
119876 le 119862(ln 119899
119899)
2119904(2119904+2120588+1)
(88)
Combining (64) (66) (69) and (88) we complete theproof of Theorem 3
Proof of Theorem 4 The proof of Theorem 4 is a direct appli-cation ofTheorem 3 under (G1)ndash(G5) the function 119902 definedby (24) satisfies (H1) see ([42] equation (2)) and (H2) (i) see([42] Lemma 6) (ii) see ([42] equation (11)) and (iii) see([30] Proof of Proposition 61) with 120588 = 120575
Proof of Theorem 5 The proof ofTheorem 5 is a consequenceofTheorem 3 under (J1)ndash(J4) the function 119902 defined by (30)satisfies (H1) and (H2) (i)-(ii) see ([31] Proposition 1) and(iii) see ([52] equation (26)) with 120588 = 120575
Proof of Theorem 6 Set V(119909) = 119891(119909)119892(119909) Following themethodology of [49] we have
119891 (119909) minus 119891 (119909) = 119878 (119909) minus 119879 (119909) (89)
10 Journal of Probability and Statistics
where
119878 (119909)
=1
119892 (119909)(V (119909) minus V (119909)
+119891 (119909) (119892 (119909) minus 119892 (119909))) 1|119892(119909)|ge119888lowast2
119879 (119909) = 119891 (119909) 1|119892(119909)|lt119888lowast2
(90)
Using (K3) and the indicator function we have
|119878 (119909)| le 119862 (|V (119909) minus V (119909)| + 1003816100381610038161003816119892 (119909) minus 119892 (119909)1003816100381610038161003816) (91)
It follows from |119892(119909)| lt 119888lowast2 cap |119892(119909)| gt 119888
lowast sube |119892(119909) minus
119892(119909)| gt 119888lowast2 (K3) (K4) and the Markov inequality that
|119879 (119909)| le 1198621|119892(119909)minus119892(119909)|gt119888lowast2
le 1198621003816100381610038161003816119892 (119909) minus 119892 (119909)
1003816100381610038161003816 (92)
The triangular inequality yields
10038161003816100381610038161003816119891 (119909) minus 119891 (119909)10038161003816100381610038161003816 le 119862 (|V (119909) minus V (119909)| + 1003816100381610038161003816119892 (119909) minus 119892 (119909)
1003816100381610038161003816) (93)
The elementary inequality (119886 + 119887)2 le 2(1198862 + 1198872) implies that
E (10038171003817100381710038171003817119891 minus 119891
100381710038171003817100381710038172
2) le 119862 (E (V minus V2
2) + E (
1003817100381710038171003817119892 minus 11989210038171003817100381710038172
2)) (94)
We now bound this two MISEs via Theorem 3
Upper Bound for the MISE of V Under (K1)ndash(K6) thefunction 119902 defined by (38) satisfies the following
(H1) With V instead of 119891 since 1205851and 119883
1are independent
with E(1205851) = 0
E (119902 (120574119895119896
1198811))
= E (1198841120574119895119896
(1198831)) = E (119891 (119883
1) 120574
119895119896(119883
1))
= int1
0
119891 (119909) 120574119895119896
(119909) 119892 (119909) 119889119909 = int1
0
V (119909) 120574119895119896
(119909) 119889119909
(95)
(H2) (i)-(ii)-(iii) with 120588 = 0
(i) since 1198841(Ω) is bounded thanks to (K2) and (K3) say
|1198841| le 119872 with 119872 gt 0 we have
sup(119909119909lowast)isin1198811(Ω)
10038161003816100381610038161003816119902 (120574119895119896
(119909 119909lowast))
10038161003816100381610038161003816
= sup(119909119909lowast)isin[minus119872119872]times[01]
10038161003816100381610038161003816119909120574119895119896(119909
lowast)10038161003816100381610038161003816
le 119872 sup119909lowastisin[01]
10038161003816100381610038161003816120574119895119896(119909
lowast)10038161003816100381610038161003816 le 11986221198952
(96)
(ii) using the boundedness of 1198841(Ω) then (K4) we have
E (10038161003816100381610038161003816119902(120574119895119896
1198811)100381610038161003816100381610038162
) = E (1198842
1(120574
119895119896(119883
1))
2
)
le 119862E ((120574119895119896
(1198831))
2
)
= 119862int1
0
(120574119895119896
(119909))2
119892 (119909) 119889119909
le 119862int1
0
(120574119895119896
(119909))2
119889119909 le 119862
(97)
(iii) using the boundedness of 1198841(Ω) and making the
change of variables 119910 = 2119895119909 minus 119896 we obtain
int1198811(Ω)
10038161003816100381610038161003816119902 (120574119895119896
119909)10038161003816100381610038161003816 119889119909
= (int119872
minus119872
|119909| 119889119909)(int1
0
10038161003816100381610038161003816120574119895119896(119909
lowast)10038161003816100381610038161003816 119889119909lowast
)
= 1198722 int1
0
10038161003816100381610038161003816120574119895119896(119909)
10038161003816100381610038161003816 119889119909 le 1198622minus1198952
(98)
We conclude by applying Lemma 2 with 120588 = 0 (K5) and (K6)imply (F1) and the previous inequality implies (F2)
Therefore assuming that V isin 119861119904
119901119903(119872) with 119903 ge 1 119901 ge 2
and 119904 isin (0119873) or 119901 isin [1 2) and 119904 isin (1119901119873) Theorem 3proves the existence of a constant 119862 gt 0 satisfying
E (V minus V22) le 119862(
ln 119899
119899)
2119904(2119904+1)
(99)
Upper Bound for theMISE of 119892 Under (K1)ndash(K6) proceedingas the previous point we show that the function 119902 defined by(39) satisfies (H1) with 119892 instead of 119891 and 119883
119905instead of 119881
119905
and(H2) (i)-(ii)-(iii) with 120588 = 0Therefore assuming that 119892 isin 119861119904
119901119903(119872) with 119903 ge 1 119901 ge 2
and 119904 isin (0119873) or 119901 isin [1 2) and 119904 isin (1119901119873) Theorem 3proves the existance of a constant 119862 gt 0 satisfying
E (1003817100381710038171003817119892 minus 119892
10038171003817100381710038172
2) le 119862(
ln 119899
119899)
2119904(2119904+1)
(100)
Combining (94) (99) and (100) we end the proof ofTheorem 6
Conflict of Interests
The author declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
The author is thankful to the reviewers for their commentswhich have helped in improving the presentation
Journal of Probability and Statistics 11
References
[1] P DoukhanMixing Properties and Examples vol 85 of LectureNotes in Statistics Springer New York NY USA 1994
[2] M Carrasco and X Chen ldquoMixing and moment properties ofvarious GARCH and stochastic volatility modelsrdquo EconometricTheory vol 18 no 1 pp 17ndash39 2002
[3] R C Bradley Introduction to Strong Mixing Conditions Volume1 2 3 Kendrick Press 2007
[4] P M Robinson ldquoNonparametric estimators for time seriesrdquoJournal of Time Series Analysis vol 4 pp 185ndash207 1983
[5] G G Roussas ldquoNonparametric estimation inmixing sequencesof random variablesrdquo Journal of Statistical Planning and Infer-ence vol 18 no 2 pp 135ndash149 1987
[6] G G Roussas ldquoNonparametric regression estimation undermixing conditionsrdquo Stochastic Processes and their Applicationsvol 36 no 1 pp 107ndash116 1990
[7] Y K Truong and C J Stone ldquoNonparametric function estima-tion involving time seriesrdquo The Annals of Statistics vol 20 pp77ndash97 1992
[8] L H Tran ldquoNonparametric function estimation for time seriesby local average estimatorsrdquoThe Annals of Statistics vol 21 pp1040ndash1057 1993
[9] E Masry ldquoMultivariate local polynomial regression for timeseries uniform strong consistency and ratesrdquo Journal of TimeSeries Analysis vol 17 no 6 pp 571ndash599 1996
[10] E Masry ldquoMultivariate regression estimation local polynomialfitting for time seriesrdquo Stochastic Processes and their Applica-tions vol 65 no 1 pp 81ndash101 1996
[11] E Masry and J Fan ldquoLocal polynomial estimation of regressionfunctions for mixing processesrdquo Scandinavian Journal of Statis-tics vol 24 no 2 pp 165ndash179 1997
[12] D Bosq Nonparametric Statistics for Stochastic Processes Esti-mation and Prediction vol 110 of Lecture Notes in StatisticsSpringer New York NY USA 1998
[13] E Liebscher ldquoEstimation of the density and the regressionfunction under mixing conditionsrdquo Statistics and Decisions vol19 no 1 pp 9ndash26 2001
[14] D L Donoho and I M Johnstone ldquoIdeal spatial adaptation bywavelet shrinkagerdquo Biometrika vol 81 no 3 pp 425ndash455 1994
[15] D L Donoho and I M Johnstone ldquoAdapting to unknownsmoothness via wavelet shrinkagerdquo Journal of the AmericanStatistical Association vol 90 pp 1200ndash1224 1995
[16] D Donoho I Johnstone G Kerkyacharian and D PicardldquoWavelet shrinkage asymptopiardquo Journal of the Royal StatisticalSociety B vol 57 pp 301ndash369 1995
[17] D L Donoho I M Johnstone G Kerkyacharian and DPicard ldquoDensity estimation by wavelet thresholdingrdquo Annals ofStatistics vol 24 no 2 pp 508ndash539 1996
[18] A Antoniadis ldquoWavelets in statistics a review (with discus-sion)rdquo Journal of the Italian Statistical Society vol 6 pp 97ndash1441997
[19] W Hardle G Kerkyacharian D Picard and A TsybakovWavelet Approximation and Statistical Applications vol 129 ofLectures Notes in Statistics Springer New York NY USA 1998
[20] F Leblanc ldquoWavelet linear density estimator for a discrete-timestochastic process 119871
119901-lossesrdquo Statistics and Probability Letters
vol 27 no 1 pp 71ndash84 1996[21] K Tribouley andG Viennet ldquo119871
119901adaptive density estimation in
a alpha-mixing frameworkrdquo Annales de lrsquoinstitut Henri PoincareB vol 34 no 2 pp 179ndash208 1998
[22] E Masry ldquoWavelet-based estimation of multivariate regressionfunctions in besov spacesrdquo Journal of Nonparametric Statisticsvol 12 no 2 pp 283ndash308 2000
[23] P N Patil and Y K Truong ldquoAsymptotics for wavelet basedestimates of piecewise smooth regression for stationary timeseriesrdquo Annals of the Institute of Statistical Mathematics vol 53no 1 pp 159ndash178 2001
[24] H Doosti M Afshari and H A Niroumand ldquoWavelets fornonparametric stochastic regression with mixing stochasticprocessrdquo Communications in Statistics vol 37 no 3 pp 373ndash385 2008
[25] H Doosti and H A Niroumand ldquoMultivariate stochasticregression estimation by wavelet methods for stationary timeseriesrdquo Pakistan Journal of Statistics vol 27 no 1 pp 37ndash462009
[26] H Doosti M S Islam Y P Chaubey and P Gora ldquoTwo-dimensional wavelets for nonlinear autoregressive models withan application in dynamical systemrdquo Italian Journal of Pure andApplied Mathematics no 27 pp 39ndash62 2010
[27] J Cai and H Liang ldquoNonlinear wavelet density estimation fortruncated and dependent observationsrdquo International Journal ofWavelets Multiresolution and Information Processing vol 9 no4 pp 587ndash609 2011
[28] S Niu and H Liang ldquoNonlinear wavelet estimation of condi-tional density under left-truncated and 120572-mixing assumptionsrdquoInternational Journal of Wavelets Multiresolution and Informa-tion Processing vol 9 no 6 pp 989ndash1023 2011
[29] F Benatia and D Yahia ldquoNonlinear wavelet regression functionestimator for censored dependent datardquo Journal Afrika Statis-tika vol 7 no 1 pp 391ndash411 2012
[30] C Chesneau ldquoOn the adaptive wavelet deconvolution of adensity for strong mixing sequencesrdquo Journal of the KoreanStatistical Society vol 41 no 4 pp 423ndash436 2012
[31] C Chesneau ldquoWavelet estimation of a density in a GARCH-type modelrdquo Communications in Statistics vol 42 no 1 pp 98ndash117 2013
[32] C Chesneau ldquoOn the adaptive wavelet estimation of a mul-tidimensional regression function under alpha-mixing depen-dence beyond the standard assumptions on the noiserdquo Com-mentationes Mathematicae Universitatis Carolinae vol 4 pp527ndash556 2013
[33] Y P Chaubey and E Shirazi ldquoOn MISE of a nonlinear waveletestimator of the regression function based on biased data understrong mixing rdquo Communications in Statistics In press
[34] M Abbaszadeh andM Emadi ldquoWavelet density estimation andstatistical evidences role for a garch model in the weighteddistributionrdquo Applied Mathematics vol 4 no 2 pp 10ndash4162013
[35] B Delyon and A Juditsky ldquoOn minimax wavelet estimatorsrdquoApplied and Computational Harmonic Analysis vol 3 no 3 pp215ndash228 1996
[36] G Kerkyacharian and D Picard ldquoThresholding algorithmsmaxisets and well-concentrated basesrdquo Test vol 9 no 2 pp283ndash344 2000
[37] I Daubechies Ten Lectures on Wavelets SIAM 1992[38] A Cohen I Daubechies and P Vial ldquoWavelets on the interval
and fast wavelet transformsrdquo Applied and Computational Har-monic Analysis vol 1 no 1 pp 54ndash81 1993
[39] S Mallat A Wavelet Tour of Signal Processing The sparse waywith contributions fromGabriel Peyralphae ElsevierAcademicPress Amsterdam The Netherlands 3rd edition 2009
12 Journal of Probability and Statistics
[40] Y Meyer Ondelettes et Opalphaerateurs Hermann ParisFrance 1990
[41] V Genon-Catalot T Jeantheau and C Laredo ldquoStochasticvolatility models as hidden Markov models and statisticalapplicationsrdquo Bernoulli vol 6 no 6 pp 1051ndash1079 2000
[42] J Fan and J Koo ldquoWavelet deconvolutionrdquo IEEE Transactionson Information Theory vol 48 no 3 pp 734ndash747 2002
[43] A B Tsybakov Introduction alphaa lrsquoestimation non-paramalphaetrique Springer New York NY USA 2004
[44] E Masry ldquoStrong consistency and rates for deconvolutionof multivariate densities of stationary processesrdquo StochasticProcesses and their Applications vol 47 no 1 pp 53ndash74 1993
[45] R Kulik ldquoNonparametric deconvolution problem for depen-dent sequencesrdquo Electronic Journal of Statistics vol 2 pp 722ndash740 2008
[46] F Comte J Dedecker and M L Taupin ldquoAdaptive densitydeconvolution with dependent inputsrdquo Mathematical Methodsof Statistics vol 17 no 2 pp 87ndash112 2008
[47] H van Zanten and P Zareba ldquoA note on wavelet densitydeconvolution for weakly dependent datardquo Statistical Inferencefor Stochastic Processes vol 11 no 2 pp 207ndash219 2008
[48] W Hardle Applied Nonparametric Regression Cambridge Uni-versity Press Cambridge UK 1990
[49] V A Vasiliev ldquoOne investigation method of a ratios typeestimatorsrdquo in Proceedings of the 16th IFAC Symposium onSystem Identialphacation pp 1ndash6 Brussels Belgium July 2012(in progress)
[50] Y Davydov ldquoThe invariance principle for stationary processesrdquoTheory of Probability and Its Applications vol 15 no 3 pp 498ndash509 1970
[51] C Chesneau ldquoWavelet estimation via block thresholding aminimax study under 119871119901-riskrdquo Statistica Sinica vol 18 no 3pp 1007ndash1024 2008
[52] C Chesneau and H Doosti ldquoWavelet linear density estimationfor a GARCH model under various dependence structuresrdquoJournal of the Iranian Statistical Society vol 11 no 1 pp 1ndash212012
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Journal of Probability and Statistics 5
43 Nonparametric Regression Model We consider thestrictly stationary stochastic process (119881
119905)119905isinZ where for any
119905 isin Z 119881119905= (119884
119905 119883
119905)
119884119905= 119891 (119883
119905) + 120585
119905 (32)
(119883119905)119905isinZ is a strictly stationary stochastic process with
unknown density 119892 (120585119905)119905isinZ is a strictly stationary centered
stochastic process and119891 is the unknown regression functionIt is supposed that 119883
119905and 120585
119905are independent for any 119905 isin Z
and (119881119905)119905isinZ is a 120572-mixing process with exponential decay
rate (see Section 31 for a precise definition) Our aim is toestimate 119891 via 119881
1 119881
119899from (119881
119905)119905isinZ Applications of this
problem can be found in Hardle [48] Wavelet methods canbe found in Patil and Truong [23] Doosti et al [24] Doostiet al [26] and Doosti and Niroumand [25]
We formulate the following assumptions
(K1) The support of 119891 and 119892 is [0 1] and 119891 and 119892 isin
L2([0 1])(K2) 120585
1(Ω) is bounded
(K3) There exists a constant 119862 gt 0 such that
sup119909isin[01]
1003816100381610038161003816119891 (119909)1003816100381610038161003816 le 119862 (33)
(K4) There exist two constants 119888lowastgt 0 and 119862 gt 0 such that
119888lowastle inf
119909isin[01]
119892 (119909) sup119909isin[01]
119892 (119909) le 119862 (34)
(K5) Let 119906 be the density of1198811We suppose that there exists
a constant 119862 gt 0 such that
sup119909isinRtimes[01]
119906 (119909) le 119862 (35)
(K6) For any 119898 isin Z let 119906(1198811 119881119898+1)
be the density of(119881
1 119881
119898+1) We suppose that there exists a constant
119862 gt 0 such that
sup119898isinZ
sup(119909119910)isin(Rtimes[01])times(Rtimes[01])
119906(1198811 119881119898+1)
(119909 119910) le 119862 (36)
We are now in the position to present the result
Theorem 6 We consider the model (32) Suppose that (K1)ndash(K6) are satisfied Let 119891 be the truncated ratio estimatorConsider
119891 (119909) =V (119909)119892 (119909)
1|119892(119909)|ge119888lowast2
(37)
where
(i) V is defined as in (7) with
119902 (120574 (119909 119909lowast)) = 119909120574 (119909
lowast) (38)
and 120588 = 0
(ii) 119892 is defined as in (7) with 119883119905instead of 119881
119905
119902 (120574 119909) = 120574 (119909) (39)
and 120588 = 0(iii) 119888
lowastis the constant defined in (K4)
Suppose that 119891119892 isin 119861119904
119901119903(119872) and 119892 isin 119861119904
119901119903(119872) with 119903 ge 1
119901 ge 2 and 119904 isin (0119873) or 119901 isin [1 2) and 119904 isin (1119901119873) Thenthere exists a constant 119862 gt 0 such that
E (10038171003817100381710038171003817119891 minus 119891
100381710038171003817100381710038172
2) le 119862(
ln 119899
119899)
2119904(2119904+1)
(40)
The estimator (37) is derived by combining the procedureof Patil and Truong [23] with the truncated approach ofVasiliev [49]
Theorem 6 completes Patil and Truong [23] in terms ofrates of convergence under the MISE over Besov balls
Remark 7 The assumption(K2) can be relaxed with anotherstrategy to the one developed in Theorem 6 Some technicalelements are given in Chesneau [32]
Conclusion Considering the weak dependent case on theobservations we prove a general result on the rate of conver-gence attains by a hard wavelet thresholding estimator underthe MISE over Besov balls This result is flexible it can beapplied for a wide class of statistical models Moreover theobtained rate of convergence is sharp it can correspond to thenear optimal one in the minimax sense for the standard iidcase Some recent results on sophisticated statistical problemsare improved Thanks to its flexibility the perspectives ofapplications of our theoretical result in other contexts arenumerous
5 Proofs
In this section 119862 denotes any constant that does not dependon 119895 119896 and 119899 Its value may change from one term to anotherand may depend on 120601 or 120595
51 Key Lemmas Let us present two lemmas which will beused in the proofs
Lemma 8 shows a sharp covariance inequality under the120572-mixing condition
Lemma 8 (see [50]) Let (119882119905)119905isinZ be a strictly stationary 120572-
mixing process withmixing coefficient 120572119898119898 ge 0 and let ℎ and
119896 be two measurable functions Let 119901 gt 0 and 119902 gt 0 satisfying1119901 + 1119902 lt 1 such that E(|ℎ(119882
1)|119901) and E(|119896(119882
1)|119902) exist
Then there exists a constant 119862 gt 0 such that1003816100381610038161003816C119900V (ℎ (119882
1) 119896 (119882
119898+1))1003816100381610038161003816
le 1198621205721minus1119901minus1119902
119898(E (
1003816100381610038161003816ℎ (1198821)1003816100381610038161003816119901
))1119901
(E(1003816100381610038161003816119896 (119882
1)1003816100381610038161003816119902
))1119902
(41)
Lemma 9 below presents a concentration inequality for120572-mixing processes
6 Journal of Probability and Statistics
Lemma9 (see [13]) Let (119882119905)119905isinZ be a strictly stationary process
with the 119898th strongly mixing coefficient 120572119898 119898 ge 0 let 119899 be a
positive integer let ℎ R rarr C be a measurable function andfor any 119905 isin Z 119880
119905= ℎ(119882
119905) We assume that E(119880
1) = 0 and
there exists a constant 119872 gt 0 satisfying |1198801| le 119872 Then for
any 119898 isin 1 [1198992] and 120582 gt 0 we have
P(1
119899
100381610038161003816100381610038161003816100381610038161003816
119899
sum119894=1
119880119894
100381610038161003816100381610038161003816100381610038161003816ge 120582)
le 4 exp(minus1205822119899
16 (119863119898119898 + 1205821198721198983)
) + 32119872
120582119899120572
119898
(42)
where
119863119898
= max119897isin12119898
V (119897
sum119894=1
119880119894) (43)
52 Intermediary Results
Proof of Lemma 2 Using a standard expression of the covari-ance and (F1) as well as(F2) we obtain
10038161003816100381610038161003816C119900V (119902 (120574119895119896
119881119898+1
) 119902 (120574119895119896
1198811))
10038161003816100381610038161003816
=100381610038161003816100381610038161003816100381610038161003816int1198811(Ω)
int1198811(Ω)
119902 (120574119895119896
119909) 119902 (120574119895119896
119910)
times (119906(1198811 119881119898+1)
(119909 119910) minus 119906 (119909) 119906 (119910)) 119889119909119889119910100381610038161003816100381610038161003816100381610038161003816
le int1198811(Ω)
int1198811(Ω)
10038161003816100381610038161003816119902 (120574119895119896
119909)1003816100381610038161003816100381610038161003816100381610038161003816119902 (120574
119895119896 119910)
10038161003816100381610038161003816
times10038161003816100381610038161003816119906(1198811 119881119898+1)
(119909 119910) minus 119906 (119909) 119906 (119910)10038161003816100381610038161003816 119889119909 119889119910
le 119862(int1198811(Ω)
10038161003816100381610038161003816119902 (120574119895119896
119909)10038161003816100381610038161003816 119889119909)
2
le 119862221205881198952minus119895
(44)
This ends the proof of Lemma 2
Proposition 10 proves probability and moments inequal-ities satisfied by the estimators (8)
Proposition 10 Let 119895119896
and 120573119895119896
be defined as in (8) under(H1) and (H2) let 119895
0be (9) and let 119895
1be (10)
(a) There exists a constant 119862 gt 0 such that for any 119895 isin119895
0 119895
1 and 119896 isin Λ
119895
E (10038161003816100381610038161003816119888119895119896 minus 119888
119895119896
100381610038161003816100381610038162
) le 11986222120588119895 1
119899 (45)
E (10038161003816100381610038161003816119889119895119896
minus 119889119895119896
100381610038161003816100381610038162
) le 11986222120588119895 1
119899 (46)
(b) There exists a constant 119862 gt 0 such that for any 119895 isin119895
0 119895
1 and 119896 isin Λ
119895
E (10038161003816100381610038161003816119889119895119896
minus 119889119895119896
100381610038161003816100381610038164
) le 11986224120588119895 (47)
(c) Let 120582119895be defined as in (11) There exists a constant 119862 gt
0 such that for any 120581 large enough 119895 isin 1198950 119895
1 and
119896 isin Λ119895 we have
P(10038161003816100381610038161003816119889119895119896
minus 119889119895119896
10038161003816100381610038161003816 ge120581120582
119895
2) le 119862
1
1198994 (48)
Proof of Proposition 10 (a) Using (H1) and the stationarity of(119881
119905)119905isinZ we obtain
E (10038161003816100381610038161003816119888119895119896 minus 119888
119895119896
100381610038161003816100381610038162
) = V (119888119895119896
)
=1
1198992
119899
sum119894=1
V (119902 (120601119895119896
119881119894)) +
2
1198992
times119899
sumV=2
Vminus1
sumℓ=1
Re (C119900V (119902 (120601
119895119896 119881V) 119902 (120601
119895119896 119881
ℓ)))
=1
1198992
119899
sum119894=1
V (119902 (120601119895k 119881119894
)) +2
1198992
times119899minus1
sum119898=1
(119899 minus 119898)Re (C119900V (119902 (120601
119895119896 119881
119898+1) 119902 (120601
119895119896 119881
1)))
le1
119899(E (
10038161003816100381610038161003816119902 (120601119895119896
1198811)100381610038161003816100381610038162
)
+2119899minus1
sum119898=1
10038161003816100381610038161003816C119900V (119902 (120601119895119896
119881119898+1
) 119902 (120601119895119896
1198811))
10038161003816100381610038161003816)
(49)
By (H2)-(ii) we get
E (10038161003816100381610038161003816119902(120601119895119896
1198811)100381610038161003816100381610038162
) le 11986222120588119895 (50)
For the covariance term note that
119899minus1
sum119898=1
10038161003816100381610038161003816C119900V (119902 (120601119895119896
119881119898+1
) 119902 (120601119895119896
1198811))
10038161003816100381610038161003816 = 119860 + 119861 (51)
where
119860 =[ln 119899120579]minus1
sum119898=1
10038161003816100381610038161003816C119900V (119902 (120601119895119896
119881119898+1
) 119902 (120601119895119896
1198811))
10038161003816100381610038161003816
119861 =119899minus1
sum119898=[(ln 119899)120579]
10038161003816100381610038161003816C119900V (119902 (120601119895119896
119881119898+1
) 119902 (120601119895119896
1198811))
10038161003816100381610038161003816
(52)
It follows from (H2)-(iii) and 2minus119895 le 2minus1198950 lt 2(ln 119899)minus1 that
119860 le 119862221205881198952minus119895 [ln 119899
120579] le 11986222120588119895 (53)
Journal of Probability and Statistics 7
The Davydov inequality described in Lemma 8 with 119901 = 119902 =
4 (H2)-(i)-(ii) and 2119895 le 21198951 le 119899 give
119861 le 119862radicE (10038161003816100381610038161003816119902(120601119895119896
1198811)100381610038161003816100381610038164
)119899minus1
sum119898=[(ln 119899)120579]
radic120572119898
le 119862212058811989521198952radicE (10038161003816100381610038161003816119902(120601119895119896
1198811)100381610038161003816100381610038162
)infin
sum119898=[(ln 119899)120579]
119890minus1205791198982
= 11986222120588119895radic119899119890minus(ln 119899)2 le 11986222120588119895
(54)
Thus119899minus1
sum119898=1
10038161003816100381610038161003816C119900V (119902 (120601119895119896
119881119898+1
) 119902 (120601119895119896
1198811))
10038161003816100381610038161003816 le 11986222120588119895 (55)
Putting (49) (50) and (55) together the first point in (a)is proved The proof of the second point is identical with 120595instead of 120601
(b) Thanks to (H2)-(i) we have |119889119895119896
| le
sup119909isin1198811(Ω)
|119902(120595119895119896
119909)| le 119862212058811989521198952 It follows from thetriangular inequality and |119889
119895119896| le 119891
2le 119862 that
10038161003816100381610038161003816119889119895119896minus 119889
119895119896
10038161003816100381610038161003816 le10038161003816100381610038161003816119889119895119896
10038161003816100381610038161003816 +10038161003816100381610038161003816119889119895119896
10038161003816100381610038161003816 le 119862212058811989521198952 (56)
This inequality and the second result of (a) yield
E (10038161003816100381610038161003816119889119895119896
minus 119889119895119896
100381610038161003816100381610038164
) le 119862221205881198952119895E (
10038161003816100381610038161003816119889119895119896minus 119889
119895119896
100381610038161003816100381610038162
) le 119862241205881198952119895 1
119899
(57)
Using 2119895 le 21198951 le 119899 the proof of (b) is completed(c) We will use the Liebscher inequality described in
Lemma 9 Let us set
119880119894= 119902 (120595
119895119896 119881
119894) minus E (119902 (120595
119895119896 119881
1)) (58)
We have E(1198801) = 0 and by(H2)-(i) and 2119895 le 21198951 le
119899(ln 119899)3
1003816100381610038161003816119880119894
1003816100381610038161003816 le 2 sup119909isin1198811(Ω)
10038161003816100381610038161003816119902 (120595119895119896
119909)10038161003816100381610038161003816 le 119862212058811989521198952 le 1198622120588119895
radic119899
(ln 119899)3
(59)
(so 119872 = 1198622120588119895radic119899(ln 119899)3)Proceeding as for the proofs of the bounds in (a) for any
integer 119897 le 119862 ln 119899 since 2minus119895 le 2minus1198950 le 2(ln 119899)minus1 we show that
V (119897
sum119894=1
119880119894)
= V (119897
sum119894=1
119902 (120595119895119896
119881119894)) le 11986222120588119895 (119897 + 11989722minus119895) le 11986222120588119895119897
(60)
Therefore
119863119898
= max119897isin12119898
V (119897
sum119894=1
119880119894) le 11986222120588119895119898 (61)
Owing to Lemma 9 applied with 1198801 119880
119899 120582 = 120581120582
1198952
119898 = [radic120581 ln 119899] 119872 = 1198622120588119895radic119899(ln 119899)3 and the bound (61) weobtain
P(10038161003816100381610038161003816119889119895119896
minus 119889119895119896
10038161003816100381610038161003816 ge120581120582
119895
2)
le 119862(exp(minus11986212058121205822
119895119899
119863119898119898 + 120581120582
119895119898119872
) +119872
120582119895
119899119890minus120579119898)
le 119862( exp (minus119862((120581222120588119895 ln 119899)
times (22120588119895 + 1205812120588119895radic (ln 119899)
119899
times [radic120581 ln 119899] 2120588119895
radic119899
(ln 119899)3)
minus1
))
+ radic119899(ln 119899)3
(ln 119899) 119899119899119890minus120579[radic120581 ln 119899])
le 119862(119899minus1198621205812(1+120581
32) + 1198992minus120579radic120581)
(62)
Taking 120581 large enough the last term is bounded by1198621198994Thiscompletes the proof of(c)
This completes the proof of Proposition 10
Proof of Theorem 3 Theorem 3 can be proved by combiningarguments of ([36] Theorem 51) and ([51] Theorem 42) Itis close to ([30] Proof of Theorem 2) by taking 120579 rarr infin Theinterested reader can find the details below
We consider the following wavelet decomposition for 119891
119891 (119909) = sum119896isinΛ 1198950
1198881198950 119896
1206011198950 119896
(119909) +infin
sum119895=1198950
sum119896isinΛ 119895
119889119895119896
120595119895119896
(119909) (63)
where 1198881198950 119896
= int1
0119891(119909)120601
1198950 119896(119909)119889119909 and 119889
119895119896= int
1
0119891(119909)120595
119895119896(119909)119889119909
Using the orthonormality of the wavelet basis B theMISE of 119891 can be expressed as
E (10038171003817100381710038171003817119891 minus 119891
100381710038171003817100381710038172
2) = 119875 + 119876 + 119877 (64)
where
119875 = sum119896isinΛ 1198950
E (100381610038161003816100381610038161198881198950 119896 minus 119888
1198950 119896
100381610038161003816100381610038162
)
119876 =
1198951
sum119895=1198950
sum119896isinΛ 119895
E(100381610038161003816100381610038161003816119889
1198951198961|119889119895119896|ge120581120582119895
minus 119889119895119896
100381610038161003816100381610038161003816
2
)
119877 =infin
sum119895=1198951+1
sum119896isinΛ 119895
1198892
119895119896
(65)
8 Journal of Probability and Statistics
Let us now investigate sharp upper bounds for 119875 119877 and119876 successively
Upper Bound for 119875 The point (a) of Proposition 10 and2119904(2119904 + 2120588 + 1) lt 1 yield
119875 le 11986221198950
119899le 119862
ln 119899
119899le 119862(
ln 119899
119899)
2119904(2119904+2120588+1)
(66)
Upper Bound for 119877(i) For 119903 ge 1 and119901 ge 2 we have119891 isin 119861119904
119901119903(119872) sube 119861119904
2infin(119872)
Using 2119904(2119904 + 2120588 + 1) lt 2119904(2120588 + 1) we obtain
119877 le 119862infin
sum119895=1198951+1
2minus2119895119904 le 119862((ln 119899)3
119899)
2119904(2120588+1)
le 119862(ln 119899
119899)
2119904(2119904+2120588+1)
(67)
(ii) For 119903 ge 1 and 119901 isin [1 2) we have 119891 isin 119861119904
119901119903(119872) sube
119861119904+12minus1119901
2infin(119872) The condition 119904 gt (2120588 + 1)119901 implies
that (119904 + 12 minus 1119901)(2120588 + 1) gt 119904(2119904 + 2120588 + 1) Thus
119877 le 119862infin
sum119895=1198951+1
2minus2119895(119904+12minus1119901)
le 119862((ln 119899)3
119899)
2(119904+12minus1119901)(2120588+1)
le 119862(ln 119899
119899)
2119904(2119904+2120588+1)
(68)
Hence for 119903 ge 1 119901 ge 2 and 119904 gt 0 or 119901 isin [1 2) and119904 gt (2120588 + 1)119901 we have
119877 le 119862(ln 119899
119899)
2119904(2119904+2120588+1)
(69)
Upper Bound for 119876 Adopting the notation119863119895119896
= 119889119895119896
minus 119889119895119896
119876 can be written as
119876 =4
sum119894=1
119876119894 (70)
where
1198761=
1198951
sum119895=1198950
sum119896isinΛ 119895
E (10038161003816100381610038161003816119863119895119896
100381610038161003816100381610038162
1|119889119895119896|ge120581120582119895 |119889119895119896|lt1205811205821198952
)
1198762=
1198951
sum119895=1198950
sum119896isinΛ 119895
E (10038161003816100381610038161003816119863119895119896
100381610038161003816100381610038162
1|119889119895119896|ge120581120582119895 |119889119895119896|ge1205811205821198952
)
1198763=
1198951
sum119895=1198950
sum119896isinΛ 119895
E (1198892
1198951198961|119889119895119896|lt120581120582119895 |119889119895119896|ge2120581120582119895
)
1198764=
1198951
sum119895=1198950
sum119896isinΛ 119895
E (1198892
1198951198961|119889119895119896|lt120581120582119895 |119889119895119896|lt2120581120582119895
)
(71)
Upper Bound for 1198761
+ 1198763 Owing to the inequalities
1|119889119895119896|lt120581120582119895 |119889119895119896|ge2120581120582119895
le 1|119895119896|gt1205811205821198952
1|119889119895119896|ge120581120582119895 |119889119895119896|lt1205811205821198952
le
1|119895119896|gt1205811205821198952
and 1|119889119895119896|lt120581120582119895 |119889119895119896|ge2120581120582119895
le 1|119889119895119896|le2|119895119896|
theCauchy-Schwarz inequality and the points(b) and (c) ofProposition 10 we have
1198761+ 119876
3le 119862
1198951
sum119895=1198950
sum119896isinΛ 119895
E (10038161003816100381610038161003816119863119895119896
100381610038161003816100381610038162
1|119895119896|gt1205811205821198952
)
le 119862
1198951
sum119895=1198950
sum119896isinΛ 119895
(E (10038161003816100381610038161003816119863119895119896
100381610038161003816100381610038164
))12
(P(10038161003816100381610038161003816119863119895119896
10038161003816100381610038161003816 gt120581120582
119895
2))
12
le 1198621
1198992
1198951
sum119895=1198950
2119895(1+2120588) le 1198621
119899le 119862(
ln 119899
119899)
2119904(2119904+2120588+1)
(72)
Upper Bound for 1198762 It follows from the point(a) of
Proposition 10 that
1198762le
1198951
sum119895=1198950
sum119896isinΛ 119895
E (10038161003816100381610038161003816119863119895119896
100381610038161003816100381610038162
) 1|119889119895119896|ge1205811205821198952
le 1198621
119899
1198951
sum119895=j0
22120588119895 sum119896isinΛ 119895
1|119889119895119896|gt1205811205821198952
(73)
Let us now introduce the integer 119895lowastdefined by
2119895lowast = [(119899
ln 119899)
1(2119904+2120588+1)
] (74)
Note that 119895lowastisin 119895
0 119895
1 for 119899 large enough
Then 1198762can be bounded as
1198762le 119876
21+ 119876
22 (75)
where
11987621
= 1198621
119899
119895lowast
sum119895=1198950
22120588119895 sum119896isinΛ 119895
1|119889119895119896|gt1205811205821198952
11987622
= 1198621
119899
1198951
sum119895=119895lowast+1
22120588119895 sum119896isinΛ 119895
1|119889119895119896|gt1205811205821198952
(76)
On the one hand we have
11987621
le 119862ln 119899
119899
119895lowast
sum119895=1198950
2119895(1+2120588) le 119862(ln 119899
119899)
2119904(2119904+2120588+1)
(77)
On the other hand we have the following
Journal of Probability and Statistics 9
(i) For 119903 ge 1 and 119901 ge 2 the Markov inequality and 119891 isin119861119904
119901119903(119872) sube 119861119904
2infin(119872) yield
11987622
le 119862ln 119899
119899
1198951
sum119895=119895lowast+1
22120588119895 1
1205822
119895
sum119896isinΛ 119895
1198892
119895119896le 119862
infin
sum119895=119895lowast+1
sum119896isinΛ 119895
1198892
119895119896
le 119862infin
sum119895=119895lowast+1
2minus2119895119904 le 119862(ln 119899
119899)
2119904(2119904+2120588+1)
(78)
(ii) For 119903 ge 1 119901 isin [1 2) and 119904 gt (2120588 + 1)119901 the Markovinequality 119891 isin 119861119904
119901119903(119872) and (2119904 + 2120588 + 1)(2 minus 119901)2 +
(119904 + 12 minus 1119901 + 120588 minus 2120588119901)119901 = 2119904 imply that
11987622
le 119862ln 119899
119899
1198951
sum119895=119895lowast+1
22120588119895 1
120582119901
119895
sum119896isinΛ 119895
10038161003816100381610038161003816119889119895119896
10038161003816100381610038161003816119901
le 119862(ln 119899
119899)
(2minus119901)2 infin
sum119895=119895lowast+1
2119895120588(2minus119901)2minus119895(119904+12minus1119901)119901
le 119862(ln 119899
119899)
(2minus119901)2
2minus119895lowast(119904+12minus1119901+120588minus2120588119901)119901
le 119862(ln 119899
119899)
2119904(2119904+2120588+1)
(79)
Therefore for 119903 ge 1 119901 ge 2 and 119904 gt 0 or 119901 isin [1 2) and119904 gt (2120588 + 1)119901 we have
1198762le C(
ln 119899
119899)
2119904(2119904+2120588+1)
(80)
Upper Bound for 1198764 We have
1198764le
1198951
sum119895=1198950
sum119896isinΛ 119895
1198892
1198951198961|119889119895119896|lt2120581120582119895
(81)
Let 119895lowastbe the integer (74) Then 119876
4can be bound as
1198764le 119876
41+ 119876
42 (82)
where
11987641
=
119895lowast
sum119895=1198950
sum119896isinΛ 119895
1198892
1198951198961|119889119895119896|lt2120581120582119895
11987642
=
1198951
sum119895=119895lowast+1
sum119896isinΛ 119895
1198892
1198951198961|119889119895119896|lt2120581120582119895
(83)
On the one hand we have
11987641
le 119862
119895lowast
sum119895=1198950
21198951205822
119895= 119862
ln 119899
119899
119895lowast
sum119895=1198950
2119895(1+2120588) le 119862(ln 119899
119899)
2119904(2119904+2120588+1)
(84)
On the other hand we have the following
(i) For 119903 ge 1 and 119901 ge 2 since 119891 isin 119861119904
119901119903(119872) sube 119861119904
2infin(119872)
we have
11987642
leinfin
sum119895=119895lowast+1
sum119896isinΛ 119895
1198892
119895119896le 119862
infin
sum119895=119895lowast+1
2minus2119895119904 le 119862(ln 119899
119899)
2119904(2119904+2120588+1)
(85)
(ii) For 119903 ge 1 119901 isin [1 2) and 119904 gt (2120588 + 1)119901 owing to theMarkov inequality 119891 isin 119861119904
119901119903(119872) and (2119904 + 2120588 + 1)(2 minus
119901)2 + (119904 + 12 minus 1119901 + 120588 minus 2120588119901)119901 = 2119904 we get
11987642
le 119862
1198951
sum119895=119895lowast+1
1205822minus119901
119895sum
119896isinΛ 119895
10038161003816100381610038161003816119889119895119896
10038161003816100381610038161003816119901
= 119862(ln 119899
119899)
(2minus119901)2 1198951
sum119895=119895lowast+1
2119895120588(2minus119901) sum119896isinΛ 119895
10038161003816100381610038161003816119889119895119896
10038161003816100381610038161003816119901
le 119862(ln 119899
119899)
(2minus119901)2 infin
sum119895=119895lowast+1
2119895120588(2minus119901)2minus119895(119904+12minus1119901)119901
le 119862(ln 119899
119899)
(2minus119901)2
2minus119895lowast(119904+12minus1119901+120588minus2120588119901)119901
le 119862(ln 119899
119899)
2119904(2119904+2120588+1)
(86)
So for 119903 ge 1 119901 ge 2 and 119904 gt 0 or 119901 isin [1 2) and 119904 gt(2120588 + 1)119901 we have
1198764le 119862(
ln 119899
119899)
2119904(2119904+2120588+1)
(87)
Putting (70) (72) (80) and (87) together for 119903 ge 1 119901 ge 2and 119904 gt 0 or 119901 isin [1 2) and 119904 gt (2120588 + 1)119901 we obtain
119876 le 119862(ln 119899
119899)
2119904(2119904+2120588+1)
(88)
Combining (64) (66) (69) and (88) we complete theproof of Theorem 3
Proof of Theorem 4 The proof of Theorem 4 is a direct appli-cation ofTheorem 3 under (G1)ndash(G5) the function 119902 definedby (24) satisfies (H1) see ([42] equation (2)) and (H2) (i) see([42] Lemma 6) (ii) see ([42] equation (11)) and (iii) see([30] Proof of Proposition 61) with 120588 = 120575
Proof of Theorem 5 The proof ofTheorem 5 is a consequenceofTheorem 3 under (J1)ndash(J4) the function 119902 defined by (30)satisfies (H1) and (H2) (i)-(ii) see ([31] Proposition 1) and(iii) see ([52] equation (26)) with 120588 = 120575
Proof of Theorem 6 Set V(119909) = 119891(119909)119892(119909) Following themethodology of [49] we have
119891 (119909) minus 119891 (119909) = 119878 (119909) minus 119879 (119909) (89)
10 Journal of Probability and Statistics
where
119878 (119909)
=1
119892 (119909)(V (119909) minus V (119909)
+119891 (119909) (119892 (119909) minus 119892 (119909))) 1|119892(119909)|ge119888lowast2
119879 (119909) = 119891 (119909) 1|119892(119909)|lt119888lowast2
(90)
Using (K3) and the indicator function we have
|119878 (119909)| le 119862 (|V (119909) minus V (119909)| + 1003816100381610038161003816119892 (119909) minus 119892 (119909)1003816100381610038161003816) (91)
It follows from |119892(119909)| lt 119888lowast2 cap |119892(119909)| gt 119888
lowast sube |119892(119909) minus
119892(119909)| gt 119888lowast2 (K3) (K4) and the Markov inequality that
|119879 (119909)| le 1198621|119892(119909)minus119892(119909)|gt119888lowast2
le 1198621003816100381610038161003816119892 (119909) minus 119892 (119909)
1003816100381610038161003816 (92)
The triangular inequality yields
10038161003816100381610038161003816119891 (119909) minus 119891 (119909)10038161003816100381610038161003816 le 119862 (|V (119909) minus V (119909)| + 1003816100381610038161003816119892 (119909) minus 119892 (119909)
1003816100381610038161003816) (93)
The elementary inequality (119886 + 119887)2 le 2(1198862 + 1198872) implies that
E (10038171003817100381710038171003817119891 minus 119891
100381710038171003817100381710038172
2) le 119862 (E (V minus V2
2) + E (
1003817100381710038171003817119892 minus 11989210038171003817100381710038172
2)) (94)
We now bound this two MISEs via Theorem 3
Upper Bound for the MISE of V Under (K1)ndash(K6) thefunction 119902 defined by (38) satisfies the following
(H1) With V instead of 119891 since 1205851and 119883
1are independent
with E(1205851) = 0
E (119902 (120574119895119896
1198811))
= E (1198841120574119895119896
(1198831)) = E (119891 (119883
1) 120574
119895119896(119883
1))
= int1
0
119891 (119909) 120574119895119896
(119909) 119892 (119909) 119889119909 = int1
0
V (119909) 120574119895119896
(119909) 119889119909
(95)
(H2) (i)-(ii)-(iii) with 120588 = 0
(i) since 1198841(Ω) is bounded thanks to (K2) and (K3) say
|1198841| le 119872 with 119872 gt 0 we have
sup(119909119909lowast)isin1198811(Ω)
10038161003816100381610038161003816119902 (120574119895119896
(119909 119909lowast))
10038161003816100381610038161003816
= sup(119909119909lowast)isin[minus119872119872]times[01]
10038161003816100381610038161003816119909120574119895119896(119909
lowast)10038161003816100381610038161003816
le 119872 sup119909lowastisin[01]
10038161003816100381610038161003816120574119895119896(119909
lowast)10038161003816100381610038161003816 le 11986221198952
(96)
(ii) using the boundedness of 1198841(Ω) then (K4) we have
E (10038161003816100381610038161003816119902(120574119895119896
1198811)100381610038161003816100381610038162
) = E (1198842
1(120574
119895119896(119883
1))
2
)
le 119862E ((120574119895119896
(1198831))
2
)
= 119862int1
0
(120574119895119896
(119909))2
119892 (119909) 119889119909
le 119862int1
0
(120574119895119896
(119909))2
119889119909 le 119862
(97)
(iii) using the boundedness of 1198841(Ω) and making the
change of variables 119910 = 2119895119909 minus 119896 we obtain
int1198811(Ω)
10038161003816100381610038161003816119902 (120574119895119896
119909)10038161003816100381610038161003816 119889119909
= (int119872
minus119872
|119909| 119889119909)(int1
0
10038161003816100381610038161003816120574119895119896(119909
lowast)10038161003816100381610038161003816 119889119909lowast
)
= 1198722 int1
0
10038161003816100381610038161003816120574119895119896(119909)
10038161003816100381610038161003816 119889119909 le 1198622minus1198952
(98)
We conclude by applying Lemma 2 with 120588 = 0 (K5) and (K6)imply (F1) and the previous inequality implies (F2)
Therefore assuming that V isin 119861119904
119901119903(119872) with 119903 ge 1 119901 ge 2
and 119904 isin (0119873) or 119901 isin [1 2) and 119904 isin (1119901119873) Theorem 3proves the existence of a constant 119862 gt 0 satisfying
E (V minus V22) le 119862(
ln 119899
119899)
2119904(2119904+1)
(99)
Upper Bound for theMISE of 119892 Under (K1)ndash(K6) proceedingas the previous point we show that the function 119902 defined by(39) satisfies (H1) with 119892 instead of 119891 and 119883
119905instead of 119881
119905
and(H2) (i)-(ii)-(iii) with 120588 = 0Therefore assuming that 119892 isin 119861119904
119901119903(119872) with 119903 ge 1 119901 ge 2
and 119904 isin (0119873) or 119901 isin [1 2) and 119904 isin (1119901119873) Theorem 3proves the existance of a constant 119862 gt 0 satisfying
E (1003817100381710038171003817119892 minus 119892
10038171003817100381710038172
2) le 119862(
ln 119899
119899)
2119904(2119904+1)
(100)
Combining (94) (99) and (100) we end the proof ofTheorem 6
Conflict of Interests
The author declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
The author is thankful to the reviewers for their commentswhich have helped in improving the presentation
Journal of Probability and Statistics 11
References
[1] P DoukhanMixing Properties and Examples vol 85 of LectureNotes in Statistics Springer New York NY USA 1994
[2] M Carrasco and X Chen ldquoMixing and moment properties ofvarious GARCH and stochastic volatility modelsrdquo EconometricTheory vol 18 no 1 pp 17ndash39 2002
[3] R C Bradley Introduction to Strong Mixing Conditions Volume1 2 3 Kendrick Press 2007
[4] P M Robinson ldquoNonparametric estimators for time seriesrdquoJournal of Time Series Analysis vol 4 pp 185ndash207 1983
[5] G G Roussas ldquoNonparametric estimation inmixing sequencesof random variablesrdquo Journal of Statistical Planning and Infer-ence vol 18 no 2 pp 135ndash149 1987
[6] G G Roussas ldquoNonparametric regression estimation undermixing conditionsrdquo Stochastic Processes and their Applicationsvol 36 no 1 pp 107ndash116 1990
[7] Y K Truong and C J Stone ldquoNonparametric function estima-tion involving time seriesrdquo The Annals of Statistics vol 20 pp77ndash97 1992
[8] L H Tran ldquoNonparametric function estimation for time seriesby local average estimatorsrdquoThe Annals of Statistics vol 21 pp1040ndash1057 1993
[9] E Masry ldquoMultivariate local polynomial regression for timeseries uniform strong consistency and ratesrdquo Journal of TimeSeries Analysis vol 17 no 6 pp 571ndash599 1996
[10] E Masry ldquoMultivariate regression estimation local polynomialfitting for time seriesrdquo Stochastic Processes and their Applica-tions vol 65 no 1 pp 81ndash101 1996
[11] E Masry and J Fan ldquoLocal polynomial estimation of regressionfunctions for mixing processesrdquo Scandinavian Journal of Statis-tics vol 24 no 2 pp 165ndash179 1997
[12] D Bosq Nonparametric Statistics for Stochastic Processes Esti-mation and Prediction vol 110 of Lecture Notes in StatisticsSpringer New York NY USA 1998
[13] E Liebscher ldquoEstimation of the density and the regressionfunction under mixing conditionsrdquo Statistics and Decisions vol19 no 1 pp 9ndash26 2001
[14] D L Donoho and I M Johnstone ldquoIdeal spatial adaptation bywavelet shrinkagerdquo Biometrika vol 81 no 3 pp 425ndash455 1994
[15] D L Donoho and I M Johnstone ldquoAdapting to unknownsmoothness via wavelet shrinkagerdquo Journal of the AmericanStatistical Association vol 90 pp 1200ndash1224 1995
[16] D Donoho I Johnstone G Kerkyacharian and D PicardldquoWavelet shrinkage asymptopiardquo Journal of the Royal StatisticalSociety B vol 57 pp 301ndash369 1995
[17] D L Donoho I M Johnstone G Kerkyacharian and DPicard ldquoDensity estimation by wavelet thresholdingrdquo Annals ofStatistics vol 24 no 2 pp 508ndash539 1996
[18] A Antoniadis ldquoWavelets in statistics a review (with discus-sion)rdquo Journal of the Italian Statistical Society vol 6 pp 97ndash1441997
[19] W Hardle G Kerkyacharian D Picard and A TsybakovWavelet Approximation and Statistical Applications vol 129 ofLectures Notes in Statistics Springer New York NY USA 1998
[20] F Leblanc ldquoWavelet linear density estimator for a discrete-timestochastic process 119871
119901-lossesrdquo Statistics and Probability Letters
vol 27 no 1 pp 71ndash84 1996[21] K Tribouley andG Viennet ldquo119871
119901adaptive density estimation in
a alpha-mixing frameworkrdquo Annales de lrsquoinstitut Henri PoincareB vol 34 no 2 pp 179ndash208 1998
[22] E Masry ldquoWavelet-based estimation of multivariate regressionfunctions in besov spacesrdquo Journal of Nonparametric Statisticsvol 12 no 2 pp 283ndash308 2000
[23] P N Patil and Y K Truong ldquoAsymptotics for wavelet basedestimates of piecewise smooth regression for stationary timeseriesrdquo Annals of the Institute of Statistical Mathematics vol 53no 1 pp 159ndash178 2001
[24] H Doosti M Afshari and H A Niroumand ldquoWavelets fornonparametric stochastic regression with mixing stochasticprocessrdquo Communications in Statistics vol 37 no 3 pp 373ndash385 2008
[25] H Doosti and H A Niroumand ldquoMultivariate stochasticregression estimation by wavelet methods for stationary timeseriesrdquo Pakistan Journal of Statistics vol 27 no 1 pp 37ndash462009
[26] H Doosti M S Islam Y P Chaubey and P Gora ldquoTwo-dimensional wavelets for nonlinear autoregressive models withan application in dynamical systemrdquo Italian Journal of Pure andApplied Mathematics no 27 pp 39ndash62 2010
[27] J Cai and H Liang ldquoNonlinear wavelet density estimation fortruncated and dependent observationsrdquo International Journal ofWavelets Multiresolution and Information Processing vol 9 no4 pp 587ndash609 2011
[28] S Niu and H Liang ldquoNonlinear wavelet estimation of condi-tional density under left-truncated and 120572-mixing assumptionsrdquoInternational Journal of Wavelets Multiresolution and Informa-tion Processing vol 9 no 6 pp 989ndash1023 2011
[29] F Benatia and D Yahia ldquoNonlinear wavelet regression functionestimator for censored dependent datardquo Journal Afrika Statis-tika vol 7 no 1 pp 391ndash411 2012
[30] C Chesneau ldquoOn the adaptive wavelet deconvolution of adensity for strong mixing sequencesrdquo Journal of the KoreanStatistical Society vol 41 no 4 pp 423ndash436 2012
[31] C Chesneau ldquoWavelet estimation of a density in a GARCH-type modelrdquo Communications in Statistics vol 42 no 1 pp 98ndash117 2013
[32] C Chesneau ldquoOn the adaptive wavelet estimation of a mul-tidimensional regression function under alpha-mixing depen-dence beyond the standard assumptions on the noiserdquo Com-mentationes Mathematicae Universitatis Carolinae vol 4 pp527ndash556 2013
[33] Y P Chaubey and E Shirazi ldquoOn MISE of a nonlinear waveletestimator of the regression function based on biased data understrong mixing rdquo Communications in Statistics In press
[34] M Abbaszadeh andM Emadi ldquoWavelet density estimation andstatistical evidences role for a garch model in the weighteddistributionrdquo Applied Mathematics vol 4 no 2 pp 10ndash4162013
[35] B Delyon and A Juditsky ldquoOn minimax wavelet estimatorsrdquoApplied and Computational Harmonic Analysis vol 3 no 3 pp215ndash228 1996
[36] G Kerkyacharian and D Picard ldquoThresholding algorithmsmaxisets and well-concentrated basesrdquo Test vol 9 no 2 pp283ndash344 2000
[37] I Daubechies Ten Lectures on Wavelets SIAM 1992[38] A Cohen I Daubechies and P Vial ldquoWavelets on the interval
and fast wavelet transformsrdquo Applied and Computational Har-monic Analysis vol 1 no 1 pp 54ndash81 1993
[39] S Mallat A Wavelet Tour of Signal Processing The sparse waywith contributions fromGabriel Peyralphae ElsevierAcademicPress Amsterdam The Netherlands 3rd edition 2009
12 Journal of Probability and Statistics
[40] Y Meyer Ondelettes et Opalphaerateurs Hermann ParisFrance 1990
[41] V Genon-Catalot T Jeantheau and C Laredo ldquoStochasticvolatility models as hidden Markov models and statisticalapplicationsrdquo Bernoulli vol 6 no 6 pp 1051ndash1079 2000
[42] J Fan and J Koo ldquoWavelet deconvolutionrdquo IEEE Transactionson Information Theory vol 48 no 3 pp 734ndash747 2002
[43] A B Tsybakov Introduction alphaa lrsquoestimation non-paramalphaetrique Springer New York NY USA 2004
[44] E Masry ldquoStrong consistency and rates for deconvolutionof multivariate densities of stationary processesrdquo StochasticProcesses and their Applications vol 47 no 1 pp 53ndash74 1993
[45] R Kulik ldquoNonparametric deconvolution problem for depen-dent sequencesrdquo Electronic Journal of Statistics vol 2 pp 722ndash740 2008
[46] F Comte J Dedecker and M L Taupin ldquoAdaptive densitydeconvolution with dependent inputsrdquo Mathematical Methodsof Statistics vol 17 no 2 pp 87ndash112 2008
[47] H van Zanten and P Zareba ldquoA note on wavelet densitydeconvolution for weakly dependent datardquo Statistical Inferencefor Stochastic Processes vol 11 no 2 pp 207ndash219 2008
[48] W Hardle Applied Nonparametric Regression Cambridge Uni-versity Press Cambridge UK 1990
[49] V A Vasiliev ldquoOne investigation method of a ratios typeestimatorsrdquo in Proceedings of the 16th IFAC Symposium onSystem Identialphacation pp 1ndash6 Brussels Belgium July 2012(in progress)
[50] Y Davydov ldquoThe invariance principle for stationary processesrdquoTheory of Probability and Its Applications vol 15 no 3 pp 498ndash509 1970
[51] C Chesneau ldquoWavelet estimation via block thresholding aminimax study under 119871119901-riskrdquo Statistica Sinica vol 18 no 3pp 1007ndash1024 2008
[52] C Chesneau and H Doosti ldquoWavelet linear density estimationfor a GARCH model under various dependence structuresrdquoJournal of the Iranian Statistical Society vol 11 no 1 pp 1ndash212012
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
6 Journal of Probability and Statistics
Lemma9 (see [13]) Let (119882119905)119905isinZ be a strictly stationary process
with the 119898th strongly mixing coefficient 120572119898 119898 ge 0 let 119899 be a
positive integer let ℎ R rarr C be a measurable function andfor any 119905 isin Z 119880
119905= ℎ(119882
119905) We assume that E(119880
1) = 0 and
there exists a constant 119872 gt 0 satisfying |1198801| le 119872 Then for
any 119898 isin 1 [1198992] and 120582 gt 0 we have
P(1
119899
100381610038161003816100381610038161003816100381610038161003816
119899
sum119894=1
119880119894
100381610038161003816100381610038161003816100381610038161003816ge 120582)
le 4 exp(minus1205822119899
16 (119863119898119898 + 1205821198721198983)
) + 32119872
120582119899120572
119898
(42)
where
119863119898
= max119897isin12119898
V (119897
sum119894=1
119880119894) (43)
52 Intermediary Results
Proof of Lemma 2 Using a standard expression of the covari-ance and (F1) as well as(F2) we obtain
10038161003816100381610038161003816C119900V (119902 (120574119895119896
119881119898+1
) 119902 (120574119895119896
1198811))
10038161003816100381610038161003816
=100381610038161003816100381610038161003816100381610038161003816int1198811(Ω)
int1198811(Ω)
119902 (120574119895119896
119909) 119902 (120574119895119896
119910)
times (119906(1198811 119881119898+1)
(119909 119910) minus 119906 (119909) 119906 (119910)) 119889119909119889119910100381610038161003816100381610038161003816100381610038161003816
le int1198811(Ω)
int1198811(Ω)
10038161003816100381610038161003816119902 (120574119895119896
119909)1003816100381610038161003816100381610038161003816100381610038161003816119902 (120574
119895119896 119910)
10038161003816100381610038161003816
times10038161003816100381610038161003816119906(1198811 119881119898+1)
(119909 119910) minus 119906 (119909) 119906 (119910)10038161003816100381610038161003816 119889119909 119889119910
le 119862(int1198811(Ω)
10038161003816100381610038161003816119902 (120574119895119896
119909)10038161003816100381610038161003816 119889119909)
2
le 119862221205881198952minus119895
(44)
This ends the proof of Lemma 2
Proposition 10 proves probability and moments inequal-ities satisfied by the estimators (8)
Proposition 10 Let 119895119896
and 120573119895119896
be defined as in (8) under(H1) and (H2) let 119895
0be (9) and let 119895
1be (10)
(a) There exists a constant 119862 gt 0 such that for any 119895 isin119895
0 119895
1 and 119896 isin Λ
119895
E (10038161003816100381610038161003816119888119895119896 minus 119888
119895119896
100381610038161003816100381610038162
) le 11986222120588119895 1
119899 (45)
E (10038161003816100381610038161003816119889119895119896
minus 119889119895119896
100381610038161003816100381610038162
) le 11986222120588119895 1
119899 (46)
(b) There exists a constant 119862 gt 0 such that for any 119895 isin119895
0 119895
1 and 119896 isin Λ
119895
E (10038161003816100381610038161003816119889119895119896
minus 119889119895119896
100381610038161003816100381610038164
) le 11986224120588119895 (47)
(c) Let 120582119895be defined as in (11) There exists a constant 119862 gt
0 such that for any 120581 large enough 119895 isin 1198950 119895
1 and
119896 isin Λ119895 we have
P(10038161003816100381610038161003816119889119895119896
minus 119889119895119896
10038161003816100381610038161003816 ge120581120582
119895
2) le 119862
1
1198994 (48)
Proof of Proposition 10 (a) Using (H1) and the stationarity of(119881
119905)119905isinZ we obtain
E (10038161003816100381610038161003816119888119895119896 minus 119888
119895119896
100381610038161003816100381610038162
) = V (119888119895119896
)
=1
1198992
119899
sum119894=1
V (119902 (120601119895119896
119881119894)) +
2
1198992
times119899
sumV=2
Vminus1
sumℓ=1
Re (C119900V (119902 (120601
119895119896 119881V) 119902 (120601
119895119896 119881
ℓ)))
=1
1198992
119899
sum119894=1
V (119902 (120601119895k 119881119894
)) +2
1198992
times119899minus1
sum119898=1
(119899 minus 119898)Re (C119900V (119902 (120601
119895119896 119881
119898+1) 119902 (120601
119895119896 119881
1)))
le1
119899(E (
10038161003816100381610038161003816119902 (120601119895119896
1198811)100381610038161003816100381610038162
)
+2119899minus1
sum119898=1
10038161003816100381610038161003816C119900V (119902 (120601119895119896
119881119898+1
) 119902 (120601119895119896
1198811))
10038161003816100381610038161003816)
(49)
By (H2)-(ii) we get
E (10038161003816100381610038161003816119902(120601119895119896
1198811)100381610038161003816100381610038162
) le 11986222120588119895 (50)
For the covariance term note that
119899minus1
sum119898=1
10038161003816100381610038161003816C119900V (119902 (120601119895119896
119881119898+1
) 119902 (120601119895119896
1198811))
10038161003816100381610038161003816 = 119860 + 119861 (51)
where
119860 =[ln 119899120579]minus1
sum119898=1
10038161003816100381610038161003816C119900V (119902 (120601119895119896
119881119898+1
) 119902 (120601119895119896
1198811))
10038161003816100381610038161003816
119861 =119899minus1
sum119898=[(ln 119899)120579]
10038161003816100381610038161003816C119900V (119902 (120601119895119896
119881119898+1
) 119902 (120601119895119896
1198811))
10038161003816100381610038161003816
(52)
It follows from (H2)-(iii) and 2minus119895 le 2minus1198950 lt 2(ln 119899)minus1 that
119860 le 119862221205881198952minus119895 [ln 119899
120579] le 11986222120588119895 (53)
Journal of Probability and Statistics 7
The Davydov inequality described in Lemma 8 with 119901 = 119902 =
4 (H2)-(i)-(ii) and 2119895 le 21198951 le 119899 give
119861 le 119862radicE (10038161003816100381610038161003816119902(120601119895119896
1198811)100381610038161003816100381610038164
)119899minus1
sum119898=[(ln 119899)120579]
radic120572119898
le 119862212058811989521198952radicE (10038161003816100381610038161003816119902(120601119895119896
1198811)100381610038161003816100381610038162
)infin
sum119898=[(ln 119899)120579]
119890minus1205791198982
= 11986222120588119895radic119899119890minus(ln 119899)2 le 11986222120588119895
(54)
Thus119899minus1
sum119898=1
10038161003816100381610038161003816C119900V (119902 (120601119895119896
119881119898+1
) 119902 (120601119895119896
1198811))
10038161003816100381610038161003816 le 11986222120588119895 (55)
Putting (49) (50) and (55) together the first point in (a)is proved The proof of the second point is identical with 120595instead of 120601
(b) Thanks to (H2)-(i) we have |119889119895119896
| le
sup119909isin1198811(Ω)
|119902(120595119895119896
119909)| le 119862212058811989521198952 It follows from thetriangular inequality and |119889
119895119896| le 119891
2le 119862 that
10038161003816100381610038161003816119889119895119896minus 119889
119895119896
10038161003816100381610038161003816 le10038161003816100381610038161003816119889119895119896
10038161003816100381610038161003816 +10038161003816100381610038161003816119889119895119896
10038161003816100381610038161003816 le 119862212058811989521198952 (56)
This inequality and the second result of (a) yield
E (10038161003816100381610038161003816119889119895119896
minus 119889119895119896
100381610038161003816100381610038164
) le 119862221205881198952119895E (
10038161003816100381610038161003816119889119895119896minus 119889
119895119896
100381610038161003816100381610038162
) le 119862241205881198952119895 1
119899
(57)
Using 2119895 le 21198951 le 119899 the proof of (b) is completed(c) We will use the Liebscher inequality described in
Lemma 9 Let us set
119880119894= 119902 (120595
119895119896 119881
119894) minus E (119902 (120595
119895119896 119881
1)) (58)
We have E(1198801) = 0 and by(H2)-(i) and 2119895 le 21198951 le
119899(ln 119899)3
1003816100381610038161003816119880119894
1003816100381610038161003816 le 2 sup119909isin1198811(Ω)
10038161003816100381610038161003816119902 (120595119895119896
119909)10038161003816100381610038161003816 le 119862212058811989521198952 le 1198622120588119895
radic119899
(ln 119899)3
(59)
(so 119872 = 1198622120588119895radic119899(ln 119899)3)Proceeding as for the proofs of the bounds in (a) for any
integer 119897 le 119862 ln 119899 since 2minus119895 le 2minus1198950 le 2(ln 119899)minus1 we show that
V (119897
sum119894=1
119880119894)
= V (119897
sum119894=1
119902 (120595119895119896
119881119894)) le 11986222120588119895 (119897 + 11989722minus119895) le 11986222120588119895119897
(60)
Therefore
119863119898
= max119897isin12119898
V (119897
sum119894=1
119880119894) le 11986222120588119895119898 (61)
Owing to Lemma 9 applied with 1198801 119880
119899 120582 = 120581120582
1198952
119898 = [radic120581 ln 119899] 119872 = 1198622120588119895radic119899(ln 119899)3 and the bound (61) weobtain
P(10038161003816100381610038161003816119889119895119896
minus 119889119895119896
10038161003816100381610038161003816 ge120581120582
119895
2)
le 119862(exp(minus11986212058121205822
119895119899
119863119898119898 + 120581120582
119895119898119872
) +119872
120582119895
119899119890minus120579119898)
le 119862( exp (minus119862((120581222120588119895 ln 119899)
times (22120588119895 + 1205812120588119895radic (ln 119899)
119899
times [radic120581 ln 119899] 2120588119895
radic119899
(ln 119899)3)
minus1
))
+ radic119899(ln 119899)3
(ln 119899) 119899119899119890minus120579[radic120581 ln 119899])
le 119862(119899minus1198621205812(1+120581
32) + 1198992minus120579radic120581)
(62)
Taking 120581 large enough the last term is bounded by1198621198994Thiscompletes the proof of(c)
This completes the proof of Proposition 10
Proof of Theorem 3 Theorem 3 can be proved by combiningarguments of ([36] Theorem 51) and ([51] Theorem 42) Itis close to ([30] Proof of Theorem 2) by taking 120579 rarr infin Theinterested reader can find the details below
We consider the following wavelet decomposition for 119891
119891 (119909) = sum119896isinΛ 1198950
1198881198950 119896
1206011198950 119896
(119909) +infin
sum119895=1198950
sum119896isinΛ 119895
119889119895119896
120595119895119896
(119909) (63)
where 1198881198950 119896
= int1
0119891(119909)120601
1198950 119896(119909)119889119909 and 119889
119895119896= int
1
0119891(119909)120595
119895119896(119909)119889119909
Using the orthonormality of the wavelet basis B theMISE of 119891 can be expressed as
E (10038171003817100381710038171003817119891 minus 119891
100381710038171003817100381710038172
2) = 119875 + 119876 + 119877 (64)
where
119875 = sum119896isinΛ 1198950
E (100381610038161003816100381610038161198881198950 119896 minus 119888
1198950 119896
100381610038161003816100381610038162
)
119876 =
1198951
sum119895=1198950
sum119896isinΛ 119895
E(100381610038161003816100381610038161003816119889
1198951198961|119889119895119896|ge120581120582119895
minus 119889119895119896
100381610038161003816100381610038161003816
2
)
119877 =infin
sum119895=1198951+1
sum119896isinΛ 119895
1198892
119895119896
(65)
8 Journal of Probability and Statistics
Let us now investigate sharp upper bounds for 119875 119877 and119876 successively
Upper Bound for 119875 The point (a) of Proposition 10 and2119904(2119904 + 2120588 + 1) lt 1 yield
119875 le 11986221198950
119899le 119862
ln 119899
119899le 119862(
ln 119899
119899)
2119904(2119904+2120588+1)
(66)
Upper Bound for 119877(i) For 119903 ge 1 and119901 ge 2 we have119891 isin 119861119904
119901119903(119872) sube 119861119904
2infin(119872)
Using 2119904(2119904 + 2120588 + 1) lt 2119904(2120588 + 1) we obtain
119877 le 119862infin
sum119895=1198951+1
2minus2119895119904 le 119862((ln 119899)3
119899)
2119904(2120588+1)
le 119862(ln 119899
119899)
2119904(2119904+2120588+1)
(67)
(ii) For 119903 ge 1 and 119901 isin [1 2) we have 119891 isin 119861119904
119901119903(119872) sube
119861119904+12minus1119901
2infin(119872) The condition 119904 gt (2120588 + 1)119901 implies
that (119904 + 12 minus 1119901)(2120588 + 1) gt 119904(2119904 + 2120588 + 1) Thus
119877 le 119862infin
sum119895=1198951+1
2minus2119895(119904+12minus1119901)
le 119862((ln 119899)3
119899)
2(119904+12minus1119901)(2120588+1)
le 119862(ln 119899
119899)
2119904(2119904+2120588+1)
(68)
Hence for 119903 ge 1 119901 ge 2 and 119904 gt 0 or 119901 isin [1 2) and119904 gt (2120588 + 1)119901 we have
119877 le 119862(ln 119899
119899)
2119904(2119904+2120588+1)
(69)
Upper Bound for 119876 Adopting the notation119863119895119896
= 119889119895119896
minus 119889119895119896
119876 can be written as
119876 =4
sum119894=1
119876119894 (70)
where
1198761=
1198951
sum119895=1198950
sum119896isinΛ 119895
E (10038161003816100381610038161003816119863119895119896
100381610038161003816100381610038162
1|119889119895119896|ge120581120582119895 |119889119895119896|lt1205811205821198952
)
1198762=
1198951
sum119895=1198950
sum119896isinΛ 119895
E (10038161003816100381610038161003816119863119895119896
100381610038161003816100381610038162
1|119889119895119896|ge120581120582119895 |119889119895119896|ge1205811205821198952
)
1198763=
1198951
sum119895=1198950
sum119896isinΛ 119895
E (1198892
1198951198961|119889119895119896|lt120581120582119895 |119889119895119896|ge2120581120582119895
)
1198764=
1198951
sum119895=1198950
sum119896isinΛ 119895
E (1198892
1198951198961|119889119895119896|lt120581120582119895 |119889119895119896|lt2120581120582119895
)
(71)
Upper Bound for 1198761
+ 1198763 Owing to the inequalities
1|119889119895119896|lt120581120582119895 |119889119895119896|ge2120581120582119895
le 1|119895119896|gt1205811205821198952
1|119889119895119896|ge120581120582119895 |119889119895119896|lt1205811205821198952
le
1|119895119896|gt1205811205821198952
and 1|119889119895119896|lt120581120582119895 |119889119895119896|ge2120581120582119895
le 1|119889119895119896|le2|119895119896|
theCauchy-Schwarz inequality and the points(b) and (c) ofProposition 10 we have
1198761+ 119876
3le 119862
1198951
sum119895=1198950
sum119896isinΛ 119895
E (10038161003816100381610038161003816119863119895119896
100381610038161003816100381610038162
1|119895119896|gt1205811205821198952
)
le 119862
1198951
sum119895=1198950
sum119896isinΛ 119895
(E (10038161003816100381610038161003816119863119895119896
100381610038161003816100381610038164
))12
(P(10038161003816100381610038161003816119863119895119896
10038161003816100381610038161003816 gt120581120582
119895
2))
12
le 1198621
1198992
1198951
sum119895=1198950
2119895(1+2120588) le 1198621
119899le 119862(
ln 119899
119899)
2119904(2119904+2120588+1)
(72)
Upper Bound for 1198762 It follows from the point(a) of
Proposition 10 that
1198762le
1198951
sum119895=1198950
sum119896isinΛ 119895
E (10038161003816100381610038161003816119863119895119896
100381610038161003816100381610038162
) 1|119889119895119896|ge1205811205821198952
le 1198621
119899
1198951
sum119895=j0
22120588119895 sum119896isinΛ 119895
1|119889119895119896|gt1205811205821198952
(73)
Let us now introduce the integer 119895lowastdefined by
2119895lowast = [(119899
ln 119899)
1(2119904+2120588+1)
] (74)
Note that 119895lowastisin 119895
0 119895
1 for 119899 large enough
Then 1198762can be bounded as
1198762le 119876
21+ 119876
22 (75)
where
11987621
= 1198621
119899
119895lowast
sum119895=1198950
22120588119895 sum119896isinΛ 119895
1|119889119895119896|gt1205811205821198952
11987622
= 1198621
119899
1198951
sum119895=119895lowast+1
22120588119895 sum119896isinΛ 119895
1|119889119895119896|gt1205811205821198952
(76)
On the one hand we have
11987621
le 119862ln 119899
119899
119895lowast
sum119895=1198950
2119895(1+2120588) le 119862(ln 119899
119899)
2119904(2119904+2120588+1)
(77)
On the other hand we have the following
Journal of Probability and Statistics 9
(i) For 119903 ge 1 and 119901 ge 2 the Markov inequality and 119891 isin119861119904
119901119903(119872) sube 119861119904
2infin(119872) yield
11987622
le 119862ln 119899
119899
1198951
sum119895=119895lowast+1
22120588119895 1
1205822
119895
sum119896isinΛ 119895
1198892
119895119896le 119862
infin
sum119895=119895lowast+1
sum119896isinΛ 119895
1198892
119895119896
le 119862infin
sum119895=119895lowast+1
2minus2119895119904 le 119862(ln 119899
119899)
2119904(2119904+2120588+1)
(78)
(ii) For 119903 ge 1 119901 isin [1 2) and 119904 gt (2120588 + 1)119901 the Markovinequality 119891 isin 119861119904
119901119903(119872) and (2119904 + 2120588 + 1)(2 minus 119901)2 +
(119904 + 12 minus 1119901 + 120588 minus 2120588119901)119901 = 2119904 imply that
11987622
le 119862ln 119899
119899
1198951
sum119895=119895lowast+1
22120588119895 1
120582119901
119895
sum119896isinΛ 119895
10038161003816100381610038161003816119889119895119896
10038161003816100381610038161003816119901
le 119862(ln 119899
119899)
(2minus119901)2 infin
sum119895=119895lowast+1
2119895120588(2minus119901)2minus119895(119904+12minus1119901)119901
le 119862(ln 119899
119899)
(2minus119901)2
2minus119895lowast(119904+12minus1119901+120588minus2120588119901)119901
le 119862(ln 119899
119899)
2119904(2119904+2120588+1)
(79)
Therefore for 119903 ge 1 119901 ge 2 and 119904 gt 0 or 119901 isin [1 2) and119904 gt (2120588 + 1)119901 we have
1198762le C(
ln 119899
119899)
2119904(2119904+2120588+1)
(80)
Upper Bound for 1198764 We have
1198764le
1198951
sum119895=1198950
sum119896isinΛ 119895
1198892
1198951198961|119889119895119896|lt2120581120582119895
(81)
Let 119895lowastbe the integer (74) Then 119876
4can be bound as
1198764le 119876
41+ 119876
42 (82)
where
11987641
=
119895lowast
sum119895=1198950
sum119896isinΛ 119895
1198892
1198951198961|119889119895119896|lt2120581120582119895
11987642
=
1198951
sum119895=119895lowast+1
sum119896isinΛ 119895
1198892
1198951198961|119889119895119896|lt2120581120582119895
(83)
On the one hand we have
11987641
le 119862
119895lowast
sum119895=1198950
21198951205822
119895= 119862
ln 119899
119899
119895lowast
sum119895=1198950
2119895(1+2120588) le 119862(ln 119899
119899)
2119904(2119904+2120588+1)
(84)
On the other hand we have the following
(i) For 119903 ge 1 and 119901 ge 2 since 119891 isin 119861119904
119901119903(119872) sube 119861119904
2infin(119872)
we have
11987642
leinfin
sum119895=119895lowast+1
sum119896isinΛ 119895
1198892
119895119896le 119862
infin
sum119895=119895lowast+1
2minus2119895119904 le 119862(ln 119899
119899)
2119904(2119904+2120588+1)
(85)
(ii) For 119903 ge 1 119901 isin [1 2) and 119904 gt (2120588 + 1)119901 owing to theMarkov inequality 119891 isin 119861119904
119901119903(119872) and (2119904 + 2120588 + 1)(2 minus
119901)2 + (119904 + 12 minus 1119901 + 120588 minus 2120588119901)119901 = 2119904 we get
11987642
le 119862
1198951
sum119895=119895lowast+1
1205822minus119901
119895sum
119896isinΛ 119895
10038161003816100381610038161003816119889119895119896
10038161003816100381610038161003816119901
= 119862(ln 119899
119899)
(2minus119901)2 1198951
sum119895=119895lowast+1
2119895120588(2minus119901) sum119896isinΛ 119895
10038161003816100381610038161003816119889119895119896
10038161003816100381610038161003816119901
le 119862(ln 119899
119899)
(2minus119901)2 infin
sum119895=119895lowast+1
2119895120588(2minus119901)2minus119895(119904+12minus1119901)119901
le 119862(ln 119899
119899)
(2minus119901)2
2minus119895lowast(119904+12minus1119901+120588minus2120588119901)119901
le 119862(ln 119899
119899)
2119904(2119904+2120588+1)
(86)
So for 119903 ge 1 119901 ge 2 and 119904 gt 0 or 119901 isin [1 2) and 119904 gt(2120588 + 1)119901 we have
1198764le 119862(
ln 119899
119899)
2119904(2119904+2120588+1)
(87)
Putting (70) (72) (80) and (87) together for 119903 ge 1 119901 ge 2and 119904 gt 0 or 119901 isin [1 2) and 119904 gt (2120588 + 1)119901 we obtain
119876 le 119862(ln 119899
119899)
2119904(2119904+2120588+1)
(88)
Combining (64) (66) (69) and (88) we complete theproof of Theorem 3
Proof of Theorem 4 The proof of Theorem 4 is a direct appli-cation ofTheorem 3 under (G1)ndash(G5) the function 119902 definedby (24) satisfies (H1) see ([42] equation (2)) and (H2) (i) see([42] Lemma 6) (ii) see ([42] equation (11)) and (iii) see([30] Proof of Proposition 61) with 120588 = 120575
Proof of Theorem 5 The proof ofTheorem 5 is a consequenceofTheorem 3 under (J1)ndash(J4) the function 119902 defined by (30)satisfies (H1) and (H2) (i)-(ii) see ([31] Proposition 1) and(iii) see ([52] equation (26)) with 120588 = 120575
Proof of Theorem 6 Set V(119909) = 119891(119909)119892(119909) Following themethodology of [49] we have
119891 (119909) minus 119891 (119909) = 119878 (119909) minus 119879 (119909) (89)
10 Journal of Probability and Statistics
where
119878 (119909)
=1
119892 (119909)(V (119909) minus V (119909)
+119891 (119909) (119892 (119909) minus 119892 (119909))) 1|119892(119909)|ge119888lowast2
119879 (119909) = 119891 (119909) 1|119892(119909)|lt119888lowast2
(90)
Using (K3) and the indicator function we have
|119878 (119909)| le 119862 (|V (119909) minus V (119909)| + 1003816100381610038161003816119892 (119909) minus 119892 (119909)1003816100381610038161003816) (91)
It follows from |119892(119909)| lt 119888lowast2 cap |119892(119909)| gt 119888
lowast sube |119892(119909) minus
119892(119909)| gt 119888lowast2 (K3) (K4) and the Markov inequality that
|119879 (119909)| le 1198621|119892(119909)minus119892(119909)|gt119888lowast2
le 1198621003816100381610038161003816119892 (119909) minus 119892 (119909)
1003816100381610038161003816 (92)
The triangular inequality yields
10038161003816100381610038161003816119891 (119909) minus 119891 (119909)10038161003816100381610038161003816 le 119862 (|V (119909) minus V (119909)| + 1003816100381610038161003816119892 (119909) minus 119892 (119909)
1003816100381610038161003816) (93)
The elementary inequality (119886 + 119887)2 le 2(1198862 + 1198872) implies that
E (10038171003817100381710038171003817119891 minus 119891
100381710038171003817100381710038172
2) le 119862 (E (V minus V2
2) + E (
1003817100381710038171003817119892 minus 11989210038171003817100381710038172
2)) (94)
We now bound this two MISEs via Theorem 3
Upper Bound for the MISE of V Under (K1)ndash(K6) thefunction 119902 defined by (38) satisfies the following
(H1) With V instead of 119891 since 1205851and 119883
1are independent
with E(1205851) = 0
E (119902 (120574119895119896
1198811))
= E (1198841120574119895119896
(1198831)) = E (119891 (119883
1) 120574
119895119896(119883
1))
= int1
0
119891 (119909) 120574119895119896
(119909) 119892 (119909) 119889119909 = int1
0
V (119909) 120574119895119896
(119909) 119889119909
(95)
(H2) (i)-(ii)-(iii) with 120588 = 0
(i) since 1198841(Ω) is bounded thanks to (K2) and (K3) say
|1198841| le 119872 with 119872 gt 0 we have
sup(119909119909lowast)isin1198811(Ω)
10038161003816100381610038161003816119902 (120574119895119896
(119909 119909lowast))
10038161003816100381610038161003816
= sup(119909119909lowast)isin[minus119872119872]times[01]
10038161003816100381610038161003816119909120574119895119896(119909
lowast)10038161003816100381610038161003816
le 119872 sup119909lowastisin[01]
10038161003816100381610038161003816120574119895119896(119909
lowast)10038161003816100381610038161003816 le 11986221198952
(96)
(ii) using the boundedness of 1198841(Ω) then (K4) we have
E (10038161003816100381610038161003816119902(120574119895119896
1198811)100381610038161003816100381610038162
) = E (1198842
1(120574
119895119896(119883
1))
2
)
le 119862E ((120574119895119896
(1198831))
2
)
= 119862int1
0
(120574119895119896
(119909))2
119892 (119909) 119889119909
le 119862int1
0
(120574119895119896
(119909))2
119889119909 le 119862
(97)
(iii) using the boundedness of 1198841(Ω) and making the
change of variables 119910 = 2119895119909 minus 119896 we obtain
int1198811(Ω)
10038161003816100381610038161003816119902 (120574119895119896
119909)10038161003816100381610038161003816 119889119909
= (int119872
minus119872
|119909| 119889119909)(int1
0
10038161003816100381610038161003816120574119895119896(119909
lowast)10038161003816100381610038161003816 119889119909lowast
)
= 1198722 int1
0
10038161003816100381610038161003816120574119895119896(119909)
10038161003816100381610038161003816 119889119909 le 1198622minus1198952
(98)
We conclude by applying Lemma 2 with 120588 = 0 (K5) and (K6)imply (F1) and the previous inequality implies (F2)
Therefore assuming that V isin 119861119904
119901119903(119872) with 119903 ge 1 119901 ge 2
and 119904 isin (0119873) or 119901 isin [1 2) and 119904 isin (1119901119873) Theorem 3proves the existence of a constant 119862 gt 0 satisfying
E (V minus V22) le 119862(
ln 119899
119899)
2119904(2119904+1)
(99)
Upper Bound for theMISE of 119892 Under (K1)ndash(K6) proceedingas the previous point we show that the function 119902 defined by(39) satisfies (H1) with 119892 instead of 119891 and 119883
119905instead of 119881
119905
and(H2) (i)-(ii)-(iii) with 120588 = 0Therefore assuming that 119892 isin 119861119904
119901119903(119872) with 119903 ge 1 119901 ge 2
and 119904 isin (0119873) or 119901 isin [1 2) and 119904 isin (1119901119873) Theorem 3proves the existance of a constant 119862 gt 0 satisfying
E (1003817100381710038171003817119892 minus 119892
10038171003817100381710038172
2) le 119862(
ln 119899
119899)
2119904(2119904+1)
(100)
Combining (94) (99) and (100) we end the proof ofTheorem 6
Conflict of Interests
The author declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
The author is thankful to the reviewers for their commentswhich have helped in improving the presentation
Journal of Probability and Statistics 11
References
[1] P DoukhanMixing Properties and Examples vol 85 of LectureNotes in Statistics Springer New York NY USA 1994
[2] M Carrasco and X Chen ldquoMixing and moment properties ofvarious GARCH and stochastic volatility modelsrdquo EconometricTheory vol 18 no 1 pp 17ndash39 2002
[3] R C Bradley Introduction to Strong Mixing Conditions Volume1 2 3 Kendrick Press 2007
[4] P M Robinson ldquoNonparametric estimators for time seriesrdquoJournal of Time Series Analysis vol 4 pp 185ndash207 1983
[5] G G Roussas ldquoNonparametric estimation inmixing sequencesof random variablesrdquo Journal of Statistical Planning and Infer-ence vol 18 no 2 pp 135ndash149 1987
[6] G G Roussas ldquoNonparametric regression estimation undermixing conditionsrdquo Stochastic Processes and their Applicationsvol 36 no 1 pp 107ndash116 1990
[7] Y K Truong and C J Stone ldquoNonparametric function estima-tion involving time seriesrdquo The Annals of Statistics vol 20 pp77ndash97 1992
[8] L H Tran ldquoNonparametric function estimation for time seriesby local average estimatorsrdquoThe Annals of Statistics vol 21 pp1040ndash1057 1993
[9] E Masry ldquoMultivariate local polynomial regression for timeseries uniform strong consistency and ratesrdquo Journal of TimeSeries Analysis vol 17 no 6 pp 571ndash599 1996
[10] E Masry ldquoMultivariate regression estimation local polynomialfitting for time seriesrdquo Stochastic Processes and their Applica-tions vol 65 no 1 pp 81ndash101 1996
[11] E Masry and J Fan ldquoLocal polynomial estimation of regressionfunctions for mixing processesrdquo Scandinavian Journal of Statis-tics vol 24 no 2 pp 165ndash179 1997
[12] D Bosq Nonparametric Statistics for Stochastic Processes Esti-mation and Prediction vol 110 of Lecture Notes in StatisticsSpringer New York NY USA 1998
[13] E Liebscher ldquoEstimation of the density and the regressionfunction under mixing conditionsrdquo Statistics and Decisions vol19 no 1 pp 9ndash26 2001
[14] D L Donoho and I M Johnstone ldquoIdeal spatial adaptation bywavelet shrinkagerdquo Biometrika vol 81 no 3 pp 425ndash455 1994
[15] D L Donoho and I M Johnstone ldquoAdapting to unknownsmoothness via wavelet shrinkagerdquo Journal of the AmericanStatistical Association vol 90 pp 1200ndash1224 1995
[16] D Donoho I Johnstone G Kerkyacharian and D PicardldquoWavelet shrinkage asymptopiardquo Journal of the Royal StatisticalSociety B vol 57 pp 301ndash369 1995
[17] D L Donoho I M Johnstone G Kerkyacharian and DPicard ldquoDensity estimation by wavelet thresholdingrdquo Annals ofStatistics vol 24 no 2 pp 508ndash539 1996
[18] A Antoniadis ldquoWavelets in statistics a review (with discus-sion)rdquo Journal of the Italian Statistical Society vol 6 pp 97ndash1441997
[19] W Hardle G Kerkyacharian D Picard and A TsybakovWavelet Approximation and Statistical Applications vol 129 ofLectures Notes in Statistics Springer New York NY USA 1998
[20] F Leblanc ldquoWavelet linear density estimator for a discrete-timestochastic process 119871
119901-lossesrdquo Statistics and Probability Letters
vol 27 no 1 pp 71ndash84 1996[21] K Tribouley andG Viennet ldquo119871
119901adaptive density estimation in
a alpha-mixing frameworkrdquo Annales de lrsquoinstitut Henri PoincareB vol 34 no 2 pp 179ndash208 1998
[22] E Masry ldquoWavelet-based estimation of multivariate regressionfunctions in besov spacesrdquo Journal of Nonparametric Statisticsvol 12 no 2 pp 283ndash308 2000
[23] P N Patil and Y K Truong ldquoAsymptotics for wavelet basedestimates of piecewise smooth regression for stationary timeseriesrdquo Annals of the Institute of Statistical Mathematics vol 53no 1 pp 159ndash178 2001
[24] H Doosti M Afshari and H A Niroumand ldquoWavelets fornonparametric stochastic regression with mixing stochasticprocessrdquo Communications in Statistics vol 37 no 3 pp 373ndash385 2008
[25] H Doosti and H A Niroumand ldquoMultivariate stochasticregression estimation by wavelet methods for stationary timeseriesrdquo Pakistan Journal of Statistics vol 27 no 1 pp 37ndash462009
[26] H Doosti M S Islam Y P Chaubey and P Gora ldquoTwo-dimensional wavelets for nonlinear autoregressive models withan application in dynamical systemrdquo Italian Journal of Pure andApplied Mathematics no 27 pp 39ndash62 2010
[27] J Cai and H Liang ldquoNonlinear wavelet density estimation fortruncated and dependent observationsrdquo International Journal ofWavelets Multiresolution and Information Processing vol 9 no4 pp 587ndash609 2011
[28] S Niu and H Liang ldquoNonlinear wavelet estimation of condi-tional density under left-truncated and 120572-mixing assumptionsrdquoInternational Journal of Wavelets Multiresolution and Informa-tion Processing vol 9 no 6 pp 989ndash1023 2011
[29] F Benatia and D Yahia ldquoNonlinear wavelet regression functionestimator for censored dependent datardquo Journal Afrika Statis-tika vol 7 no 1 pp 391ndash411 2012
[30] C Chesneau ldquoOn the adaptive wavelet deconvolution of adensity for strong mixing sequencesrdquo Journal of the KoreanStatistical Society vol 41 no 4 pp 423ndash436 2012
[31] C Chesneau ldquoWavelet estimation of a density in a GARCH-type modelrdquo Communications in Statistics vol 42 no 1 pp 98ndash117 2013
[32] C Chesneau ldquoOn the adaptive wavelet estimation of a mul-tidimensional regression function under alpha-mixing depen-dence beyond the standard assumptions on the noiserdquo Com-mentationes Mathematicae Universitatis Carolinae vol 4 pp527ndash556 2013
[33] Y P Chaubey and E Shirazi ldquoOn MISE of a nonlinear waveletestimator of the regression function based on biased data understrong mixing rdquo Communications in Statistics In press
[34] M Abbaszadeh andM Emadi ldquoWavelet density estimation andstatistical evidences role for a garch model in the weighteddistributionrdquo Applied Mathematics vol 4 no 2 pp 10ndash4162013
[35] B Delyon and A Juditsky ldquoOn minimax wavelet estimatorsrdquoApplied and Computational Harmonic Analysis vol 3 no 3 pp215ndash228 1996
[36] G Kerkyacharian and D Picard ldquoThresholding algorithmsmaxisets and well-concentrated basesrdquo Test vol 9 no 2 pp283ndash344 2000
[37] I Daubechies Ten Lectures on Wavelets SIAM 1992[38] A Cohen I Daubechies and P Vial ldquoWavelets on the interval
and fast wavelet transformsrdquo Applied and Computational Har-monic Analysis vol 1 no 1 pp 54ndash81 1993
[39] S Mallat A Wavelet Tour of Signal Processing The sparse waywith contributions fromGabriel Peyralphae ElsevierAcademicPress Amsterdam The Netherlands 3rd edition 2009
12 Journal of Probability and Statistics
[40] Y Meyer Ondelettes et Opalphaerateurs Hermann ParisFrance 1990
[41] V Genon-Catalot T Jeantheau and C Laredo ldquoStochasticvolatility models as hidden Markov models and statisticalapplicationsrdquo Bernoulli vol 6 no 6 pp 1051ndash1079 2000
[42] J Fan and J Koo ldquoWavelet deconvolutionrdquo IEEE Transactionson Information Theory vol 48 no 3 pp 734ndash747 2002
[43] A B Tsybakov Introduction alphaa lrsquoestimation non-paramalphaetrique Springer New York NY USA 2004
[44] E Masry ldquoStrong consistency and rates for deconvolutionof multivariate densities of stationary processesrdquo StochasticProcesses and their Applications vol 47 no 1 pp 53ndash74 1993
[45] R Kulik ldquoNonparametric deconvolution problem for depen-dent sequencesrdquo Electronic Journal of Statistics vol 2 pp 722ndash740 2008
[46] F Comte J Dedecker and M L Taupin ldquoAdaptive densitydeconvolution with dependent inputsrdquo Mathematical Methodsof Statistics vol 17 no 2 pp 87ndash112 2008
[47] H van Zanten and P Zareba ldquoA note on wavelet densitydeconvolution for weakly dependent datardquo Statistical Inferencefor Stochastic Processes vol 11 no 2 pp 207ndash219 2008
[48] W Hardle Applied Nonparametric Regression Cambridge Uni-versity Press Cambridge UK 1990
[49] V A Vasiliev ldquoOne investigation method of a ratios typeestimatorsrdquo in Proceedings of the 16th IFAC Symposium onSystem Identialphacation pp 1ndash6 Brussels Belgium July 2012(in progress)
[50] Y Davydov ldquoThe invariance principle for stationary processesrdquoTheory of Probability and Its Applications vol 15 no 3 pp 498ndash509 1970
[51] C Chesneau ldquoWavelet estimation via block thresholding aminimax study under 119871119901-riskrdquo Statistica Sinica vol 18 no 3pp 1007ndash1024 2008
[52] C Chesneau and H Doosti ldquoWavelet linear density estimationfor a GARCH model under various dependence structuresrdquoJournal of the Iranian Statistical Society vol 11 no 1 pp 1ndash212012
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Journal of Probability and Statistics 7
The Davydov inequality described in Lemma 8 with 119901 = 119902 =
4 (H2)-(i)-(ii) and 2119895 le 21198951 le 119899 give
119861 le 119862radicE (10038161003816100381610038161003816119902(120601119895119896
1198811)100381610038161003816100381610038164
)119899minus1
sum119898=[(ln 119899)120579]
radic120572119898
le 119862212058811989521198952radicE (10038161003816100381610038161003816119902(120601119895119896
1198811)100381610038161003816100381610038162
)infin
sum119898=[(ln 119899)120579]
119890minus1205791198982
= 11986222120588119895radic119899119890minus(ln 119899)2 le 11986222120588119895
(54)
Thus119899minus1
sum119898=1
10038161003816100381610038161003816C119900V (119902 (120601119895119896
119881119898+1
) 119902 (120601119895119896
1198811))
10038161003816100381610038161003816 le 11986222120588119895 (55)
Putting (49) (50) and (55) together the first point in (a)is proved The proof of the second point is identical with 120595instead of 120601
(b) Thanks to (H2)-(i) we have |119889119895119896
| le
sup119909isin1198811(Ω)
|119902(120595119895119896
119909)| le 119862212058811989521198952 It follows from thetriangular inequality and |119889
119895119896| le 119891
2le 119862 that
10038161003816100381610038161003816119889119895119896minus 119889
119895119896
10038161003816100381610038161003816 le10038161003816100381610038161003816119889119895119896
10038161003816100381610038161003816 +10038161003816100381610038161003816119889119895119896
10038161003816100381610038161003816 le 119862212058811989521198952 (56)
This inequality and the second result of (a) yield
E (10038161003816100381610038161003816119889119895119896
minus 119889119895119896
100381610038161003816100381610038164
) le 119862221205881198952119895E (
10038161003816100381610038161003816119889119895119896minus 119889
119895119896
100381610038161003816100381610038162
) le 119862241205881198952119895 1
119899
(57)
Using 2119895 le 21198951 le 119899 the proof of (b) is completed(c) We will use the Liebscher inequality described in
Lemma 9 Let us set
119880119894= 119902 (120595
119895119896 119881
119894) minus E (119902 (120595
119895119896 119881
1)) (58)
We have E(1198801) = 0 and by(H2)-(i) and 2119895 le 21198951 le
119899(ln 119899)3
1003816100381610038161003816119880119894
1003816100381610038161003816 le 2 sup119909isin1198811(Ω)
10038161003816100381610038161003816119902 (120595119895119896
119909)10038161003816100381610038161003816 le 119862212058811989521198952 le 1198622120588119895
radic119899
(ln 119899)3
(59)
(so 119872 = 1198622120588119895radic119899(ln 119899)3)Proceeding as for the proofs of the bounds in (a) for any
integer 119897 le 119862 ln 119899 since 2minus119895 le 2minus1198950 le 2(ln 119899)minus1 we show that
V (119897
sum119894=1
119880119894)
= V (119897
sum119894=1
119902 (120595119895119896
119881119894)) le 11986222120588119895 (119897 + 11989722minus119895) le 11986222120588119895119897
(60)
Therefore
119863119898
= max119897isin12119898
V (119897
sum119894=1
119880119894) le 11986222120588119895119898 (61)
Owing to Lemma 9 applied with 1198801 119880
119899 120582 = 120581120582
1198952
119898 = [radic120581 ln 119899] 119872 = 1198622120588119895radic119899(ln 119899)3 and the bound (61) weobtain
P(10038161003816100381610038161003816119889119895119896
minus 119889119895119896
10038161003816100381610038161003816 ge120581120582
119895
2)
le 119862(exp(minus11986212058121205822
119895119899
119863119898119898 + 120581120582
119895119898119872
) +119872
120582119895
119899119890minus120579119898)
le 119862( exp (minus119862((120581222120588119895 ln 119899)
times (22120588119895 + 1205812120588119895radic (ln 119899)
119899
times [radic120581 ln 119899] 2120588119895
radic119899
(ln 119899)3)
minus1
))
+ radic119899(ln 119899)3
(ln 119899) 119899119899119890minus120579[radic120581 ln 119899])
le 119862(119899minus1198621205812(1+120581
32) + 1198992minus120579radic120581)
(62)
Taking 120581 large enough the last term is bounded by1198621198994Thiscompletes the proof of(c)
This completes the proof of Proposition 10
Proof of Theorem 3 Theorem 3 can be proved by combiningarguments of ([36] Theorem 51) and ([51] Theorem 42) Itis close to ([30] Proof of Theorem 2) by taking 120579 rarr infin Theinterested reader can find the details below
We consider the following wavelet decomposition for 119891
119891 (119909) = sum119896isinΛ 1198950
1198881198950 119896
1206011198950 119896
(119909) +infin
sum119895=1198950
sum119896isinΛ 119895
119889119895119896
120595119895119896
(119909) (63)
where 1198881198950 119896
= int1
0119891(119909)120601
1198950 119896(119909)119889119909 and 119889
119895119896= int
1
0119891(119909)120595
119895119896(119909)119889119909
Using the orthonormality of the wavelet basis B theMISE of 119891 can be expressed as
E (10038171003817100381710038171003817119891 minus 119891
100381710038171003817100381710038172
2) = 119875 + 119876 + 119877 (64)
where
119875 = sum119896isinΛ 1198950
E (100381610038161003816100381610038161198881198950 119896 minus 119888
1198950 119896
100381610038161003816100381610038162
)
119876 =
1198951
sum119895=1198950
sum119896isinΛ 119895
E(100381610038161003816100381610038161003816119889
1198951198961|119889119895119896|ge120581120582119895
minus 119889119895119896
100381610038161003816100381610038161003816
2
)
119877 =infin
sum119895=1198951+1
sum119896isinΛ 119895
1198892
119895119896
(65)
8 Journal of Probability and Statistics
Let us now investigate sharp upper bounds for 119875 119877 and119876 successively
Upper Bound for 119875 The point (a) of Proposition 10 and2119904(2119904 + 2120588 + 1) lt 1 yield
119875 le 11986221198950
119899le 119862
ln 119899
119899le 119862(
ln 119899
119899)
2119904(2119904+2120588+1)
(66)
Upper Bound for 119877(i) For 119903 ge 1 and119901 ge 2 we have119891 isin 119861119904
119901119903(119872) sube 119861119904
2infin(119872)
Using 2119904(2119904 + 2120588 + 1) lt 2119904(2120588 + 1) we obtain
119877 le 119862infin
sum119895=1198951+1
2minus2119895119904 le 119862((ln 119899)3
119899)
2119904(2120588+1)
le 119862(ln 119899
119899)
2119904(2119904+2120588+1)
(67)
(ii) For 119903 ge 1 and 119901 isin [1 2) we have 119891 isin 119861119904
119901119903(119872) sube
119861119904+12minus1119901
2infin(119872) The condition 119904 gt (2120588 + 1)119901 implies
that (119904 + 12 minus 1119901)(2120588 + 1) gt 119904(2119904 + 2120588 + 1) Thus
119877 le 119862infin
sum119895=1198951+1
2minus2119895(119904+12minus1119901)
le 119862((ln 119899)3
119899)
2(119904+12minus1119901)(2120588+1)
le 119862(ln 119899
119899)
2119904(2119904+2120588+1)
(68)
Hence for 119903 ge 1 119901 ge 2 and 119904 gt 0 or 119901 isin [1 2) and119904 gt (2120588 + 1)119901 we have
119877 le 119862(ln 119899
119899)
2119904(2119904+2120588+1)
(69)
Upper Bound for 119876 Adopting the notation119863119895119896
= 119889119895119896
minus 119889119895119896
119876 can be written as
119876 =4
sum119894=1
119876119894 (70)
where
1198761=
1198951
sum119895=1198950
sum119896isinΛ 119895
E (10038161003816100381610038161003816119863119895119896
100381610038161003816100381610038162
1|119889119895119896|ge120581120582119895 |119889119895119896|lt1205811205821198952
)
1198762=
1198951
sum119895=1198950
sum119896isinΛ 119895
E (10038161003816100381610038161003816119863119895119896
100381610038161003816100381610038162
1|119889119895119896|ge120581120582119895 |119889119895119896|ge1205811205821198952
)
1198763=
1198951
sum119895=1198950
sum119896isinΛ 119895
E (1198892
1198951198961|119889119895119896|lt120581120582119895 |119889119895119896|ge2120581120582119895
)
1198764=
1198951
sum119895=1198950
sum119896isinΛ 119895
E (1198892
1198951198961|119889119895119896|lt120581120582119895 |119889119895119896|lt2120581120582119895
)
(71)
Upper Bound for 1198761
+ 1198763 Owing to the inequalities
1|119889119895119896|lt120581120582119895 |119889119895119896|ge2120581120582119895
le 1|119895119896|gt1205811205821198952
1|119889119895119896|ge120581120582119895 |119889119895119896|lt1205811205821198952
le
1|119895119896|gt1205811205821198952
and 1|119889119895119896|lt120581120582119895 |119889119895119896|ge2120581120582119895
le 1|119889119895119896|le2|119895119896|
theCauchy-Schwarz inequality and the points(b) and (c) ofProposition 10 we have
1198761+ 119876
3le 119862
1198951
sum119895=1198950
sum119896isinΛ 119895
E (10038161003816100381610038161003816119863119895119896
100381610038161003816100381610038162
1|119895119896|gt1205811205821198952
)
le 119862
1198951
sum119895=1198950
sum119896isinΛ 119895
(E (10038161003816100381610038161003816119863119895119896
100381610038161003816100381610038164
))12
(P(10038161003816100381610038161003816119863119895119896
10038161003816100381610038161003816 gt120581120582
119895
2))
12
le 1198621
1198992
1198951
sum119895=1198950
2119895(1+2120588) le 1198621
119899le 119862(
ln 119899
119899)
2119904(2119904+2120588+1)
(72)
Upper Bound for 1198762 It follows from the point(a) of
Proposition 10 that
1198762le
1198951
sum119895=1198950
sum119896isinΛ 119895
E (10038161003816100381610038161003816119863119895119896
100381610038161003816100381610038162
) 1|119889119895119896|ge1205811205821198952
le 1198621
119899
1198951
sum119895=j0
22120588119895 sum119896isinΛ 119895
1|119889119895119896|gt1205811205821198952
(73)
Let us now introduce the integer 119895lowastdefined by
2119895lowast = [(119899
ln 119899)
1(2119904+2120588+1)
] (74)
Note that 119895lowastisin 119895
0 119895
1 for 119899 large enough
Then 1198762can be bounded as
1198762le 119876
21+ 119876
22 (75)
where
11987621
= 1198621
119899
119895lowast
sum119895=1198950
22120588119895 sum119896isinΛ 119895
1|119889119895119896|gt1205811205821198952
11987622
= 1198621
119899
1198951
sum119895=119895lowast+1
22120588119895 sum119896isinΛ 119895
1|119889119895119896|gt1205811205821198952
(76)
On the one hand we have
11987621
le 119862ln 119899
119899
119895lowast
sum119895=1198950
2119895(1+2120588) le 119862(ln 119899
119899)
2119904(2119904+2120588+1)
(77)
On the other hand we have the following
Journal of Probability and Statistics 9
(i) For 119903 ge 1 and 119901 ge 2 the Markov inequality and 119891 isin119861119904
119901119903(119872) sube 119861119904
2infin(119872) yield
11987622
le 119862ln 119899
119899
1198951
sum119895=119895lowast+1
22120588119895 1
1205822
119895
sum119896isinΛ 119895
1198892
119895119896le 119862
infin
sum119895=119895lowast+1
sum119896isinΛ 119895
1198892
119895119896
le 119862infin
sum119895=119895lowast+1
2minus2119895119904 le 119862(ln 119899
119899)
2119904(2119904+2120588+1)
(78)
(ii) For 119903 ge 1 119901 isin [1 2) and 119904 gt (2120588 + 1)119901 the Markovinequality 119891 isin 119861119904
119901119903(119872) and (2119904 + 2120588 + 1)(2 minus 119901)2 +
(119904 + 12 minus 1119901 + 120588 minus 2120588119901)119901 = 2119904 imply that
11987622
le 119862ln 119899
119899
1198951
sum119895=119895lowast+1
22120588119895 1
120582119901
119895
sum119896isinΛ 119895
10038161003816100381610038161003816119889119895119896
10038161003816100381610038161003816119901
le 119862(ln 119899
119899)
(2minus119901)2 infin
sum119895=119895lowast+1
2119895120588(2minus119901)2minus119895(119904+12minus1119901)119901
le 119862(ln 119899
119899)
(2minus119901)2
2minus119895lowast(119904+12minus1119901+120588minus2120588119901)119901
le 119862(ln 119899
119899)
2119904(2119904+2120588+1)
(79)
Therefore for 119903 ge 1 119901 ge 2 and 119904 gt 0 or 119901 isin [1 2) and119904 gt (2120588 + 1)119901 we have
1198762le C(
ln 119899
119899)
2119904(2119904+2120588+1)
(80)
Upper Bound for 1198764 We have
1198764le
1198951
sum119895=1198950
sum119896isinΛ 119895
1198892
1198951198961|119889119895119896|lt2120581120582119895
(81)
Let 119895lowastbe the integer (74) Then 119876
4can be bound as
1198764le 119876
41+ 119876
42 (82)
where
11987641
=
119895lowast
sum119895=1198950
sum119896isinΛ 119895
1198892
1198951198961|119889119895119896|lt2120581120582119895
11987642
=
1198951
sum119895=119895lowast+1
sum119896isinΛ 119895
1198892
1198951198961|119889119895119896|lt2120581120582119895
(83)
On the one hand we have
11987641
le 119862
119895lowast
sum119895=1198950
21198951205822
119895= 119862
ln 119899
119899
119895lowast
sum119895=1198950
2119895(1+2120588) le 119862(ln 119899
119899)
2119904(2119904+2120588+1)
(84)
On the other hand we have the following
(i) For 119903 ge 1 and 119901 ge 2 since 119891 isin 119861119904
119901119903(119872) sube 119861119904
2infin(119872)
we have
11987642
leinfin
sum119895=119895lowast+1
sum119896isinΛ 119895
1198892
119895119896le 119862
infin
sum119895=119895lowast+1
2minus2119895119904 le 119862(ln 119899
119899)
2119904(2119904+2120588+1)
(85)
(ii) For 119903 ge 1 119901 isin [1 2) and 119904 gt (2120588 + 1)119901 owing to theMarkov inequality 119891 isin 119861119904
119901119903(119872) and (2119904 + 2120588 + 1)(2 minus
119901)2 + (119904 + 12 minus 1119901 + 120588 minus 2120588119901)119901 = 2119904 we get
11987642
le 119862
1198951
sum119895=119895lowast+1
1205822minus119901
119895sum
119896isinΛ 119895
10038161003816100381610038161003816119889119895119896
10038161003816100381610038161003816119901
= 119862(ln 119899
119899)
(2minus119901)2 1198951
sum119895=119895lowast+1
2119895120588(2minus119901) sum119896isinΛ 119895
10038161003816100381610038161003816119889119895119896
10038161003816100381610038161003816119901
le 119862(ln 119899
119899)
(2minus119901)2 infin
sum119895=119895lowast+1
2119895120588(2minus119901)2minus119895(119904+12minus1119901)119901
le 119862(ln 119899
119899)
(2minus119901)2
2minus119895lowast(119904+12minus1119901+120588minus2120588119901)119901
le 119862(ln 119899
119899)
2119904(2119904+2120588+1)
(86)
So for 119903 ge 1 119901 ge 2 and 119904 gt 0 or 119901 isin [1 2) and 119904 gt(2120588 + 1)119901 we have
1198764le 119862(
ln 119899
119899)
2119904(2119904+2120588+1)
(87)
Putting (70) (72) (80) and (87) together for 119903 ge 1 119901 ge 2and 119904 gt 0 or 119901 isin [1 2) and 119904 gt (2120588 + 1)119901 we obtain
119876 le 119862(ln 119899
119899)
2119904(2119904+2120588+1)
(88)
Combining (64) (66) (69) and (88) we complete theproof of Theorem 3
Proof of Theorem 4 The proof of Theorem 4 is a direct appli-cation ofTheorem 3 under (G1)ndash(G5) the function 119902 definedby (24) satisfies (H1) see ([42] equation (2)) and (H2) (i) see([42] Lemma 6) (ii) see ([42] equation (11)) and (iii) see([30] Proof of Proposition 61) with 120588 = 120575
Proof of Theorem 5 The proof ofTheorem 5 is a consequenceofTheorem 3 under (J1)ndash(J4) the function 119902 defined by (30)satisfies (H1) and (H2) (i)-(ii) see ([31] Proposition 1) and(iii) see ([52] equation (26)) with 120588 = 120575
Proof of Theorem 6 Set V(119909) = 119891(119909)119892(119909) Following themethodology of [49] we have
119891 (119909) minus 119891 (119909) = 119878 (119909) minus 119879 (119909) (89)
10 Journal of Probability and Statistics
where
119878 (119909)
=1
119892 (119909)(V (119909) minus V (119909)
+119891 (119909) (119892 (119909) minus 119892 (119909))) 1|119892(119909)|ge119888lowast2
119879 (119909) = 119891 (119909) 1|119892(119909)|lt119888lowast2
(90)
Using (K3) and the indicator function we have
|119878 (119909)| le 119862 (|V (119909) minus V (119909)| + 1003816100381610038161003816119892 (119909) minus 119892 (119909)1003816100381610038161003816) (91)
It follows from |119892(119909)| lt 119888lowast2 cap |119892(119909)| gt 119888
lowast sube |119892(119909) minus
119892(119909)| gt 119888lowast2 (K3) (K4) and the Markov inequality that
|119879 (119909)| le 1198621|119892(119909)minus119892(119909)|gt119888lowast2
le 1198621003816100381610038161003816119892 (119909) minus 119892 (119909)
1003816100381610038161003816 (92)
The triangular inequality yields
10038161003816100381610038161003816119891 (119909) minus 119891 (119909)10038161003816100381610038161003816 le 119862 (|V (119909) minus V (119909)| + 1003816100381610038161003816119892 (119909) minus 119892 (119909)
1003816100381610038161003816) (93)
The elementary inequality (119886 + 119887)2 le 2(1198862 + 1198872) implies that
E (10038171003817100381710038171003817119891 minus 119891
100381710038171003817100381710038172
2) le 119862 (E (V minus V2
2) + E (
1003817100381710038171003817119892 minus 11989210038171003817100381710038172
2)) (94)
We now bound this two MISEs via Theorem 3
Upper Bound for the MISE of V Under (K1)ndash(K6) thefunction 119902 defined by (38) satisfies the following
(H1) With V instead of 119891 since 1205851and 119883
1are independent
with E(1205851) = 0
E (119902 (120574119895119896
1198811))
= E (1198841120574119895119896
(1198831)) = E (119891 (119883
1) 120574
119895119896(119883
1))
= int1
0
119891 (119909) 120574119895119896
(119909) 119892 (119909) 119889119909 = int1
0
V (119909) 120574119895119896
(119909) 119889119909
(95)
(H2) (i)-(ii)-(iii) with 120588 = 0
(i) since 1198841(Ω) is bounded thanks to (K2) and (K3) say
|1198841| le 119872 with 119872 gt 0 we have
sup(119909119909lowast)isin1198811(Ω)
10038161003816100381610038161003816119902 (120574119895119896
(119909 119909lowast))
10038161003816100381610038161003816
= sup(119909119909lowast)isin[minus119872119872]times[01]
10038161003816100381610038161003816119909120574119895119896(119909
lowast)10038161003816100381610038161003816
le 119872 sup119909lowastisin[01]
10038161003816100381610038161003816120574119895119896(119909
lowast)10038161003816100381610038161003816 le 11986221198952
(96)
(ii) using the boundedness of 1198841(Ω) then (K4) we have
E (10038161003816100381610038161003816119902(120574119895119896
1198811)100381610038161003816100381610038162
) = E (1198842
1(120574
119895119896(119883
1))
2
)
le 119862E ((120574119895119896
(1198831))
2
)
= 119862int1
0
(120574119895119896
(119909))2
119892 (119909) 119889119909
le 119862int1
0
(120574119895119896
(119909))2
119889119909 le 119862
(97)
(iii) using the boundedness of 1198841(Ω) and making the
change of variables 119910 = 2119895119909 minus 119896 we obtain
int1198811(Ω)
10038161003816100381610038161003816119902 (120574119895119896
119909)10038161003816100381610038161003816 119889119909
= (int119872
minus119872
|119909| 119889119909)(int1
0
10038161003816100381610038161003816120574119895119896(119909
lowast)10038161003816100381610038161003816 119889119909lowast
)
= 1198722 int1
0
10038161003816100381610038161003816120574119895119896(119909)
10038161003816100381610038161003816 119889119909 le 1198622minus1198952
(98)
We conclude by applying Lemma 2 with 120588 = 0 (K5) and (K6)imply (F1) and the previous inequality implies (F2)
Therefore assuming that V isin 119861119904
119901119903(119872) with 119903 ge 1 119901 ge 2
and 119904 isin (0119873) or 119901 isin [1 2) and 119904 isin (1119901119873) Theorem 3proves the existence of a constant 119862 gt 0 satisfying
E (V minus V22) le 119862(
ln 119899
119899)
2119904(2119904+1)
(99)
Upper Bound for theMISE of 119892 Under (K1)ndash(K6) proceedingas the previous point we show that the function 119902 defined by(39) satisfies (H1) with 119892 instead of 119891 and 119883
119905instead of 119881
119905
and(H2) (i)-(ii)-(iii) with 120588 = 0Therefore assuming that 119892 isin 119861119904
119901119903(119872) with 119903 ge 1 119901 ge 2
and 119904 isin (0119873) or 119901 isin [1 2) and 119904 isin (1119901119873) Theorem 3proves the existance of a constant 119862 gt 0 satisfying
E (1003817100381710038171003817119892 minus 119892
10038171003817100381710038172
2) le 119862(
ln 119899
119899)
2119904(2119904+1)
(100)
Combining (94) (99) and (100) we end the proof ofTheorem 6
Conflict of Interests
The author declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
The author is thankful to the reviewers for their commentswhich have helped in improving the presentation
Journal of Probability and Statistics 11
References
[1] P DoukhanMixing Properties and Examples vol 85 of LectureNotes in Statistics Springer New York NY USA 1994
[2] M Carrasco and X Chen ldquoMixing and moment properties ofvarious GARCH and stochastic volatility modelsrdquo EconometricTheory vol 18 no 1 pp 17ndash39 2002
[3] R C Bradley Introduction to Strong Mixing Conditions Volume1 2 3 Kendrick Press 2007
[4] P M Robinson ldquoNonparametric estimators for time seriesrdquoJournal of Time Series Analysis vol 4 pp 185ndash207 1983
[5] G G Roussas ldquoNonparametric estimation inmixing sequencesof random variablesrdquo Journal of Statistical Planning and Infer-ence vol 18 no 2 pp 135ndash149 1987
[6] G G Roussas ldquoNonparametric regression estimation undermixing conditionsrdquo Stochastic Processes and their Applicationsvol 36 no 1 pp 107ndash116 1990
[7] Y K Truong and C J Stone ldquoNonparametric function estima-tion involving time seriesrdquo The Annals of Statistics vol 20 pp77ndash97 1992
[8] L H Tran ldquoNonparametric function estimation for time seriesby local average estimatorsrdquoThe Annals of Statistics vol 21 pp1040ndash1057 1993
[9] E Masry ldquoMultivariate local polynomial regression for timeseries uniform strong consistency and ratesrdquo Journal of TimeSeries Analysis vol 17 no 6 pp 571ndash599 1996
[10] E Masry ldquoMultivariate regression estimation local polynomialfitting for time seriesrdquo Stochastic Processes and their Applica-tions vol 65 no 1 pp 81ndash101 1996
[11] E Masry and J Fan ldquoLocal polynomial estimation of regressionfunctions for mixing processesrdquo Scandinavian Journal of Statis-tics vol 24 no 2 pp 165ndash179 1997
[12] D Bosq Nonparametric Statistics for Stochastic Processes Esti-mation and Prediction vol 110 of Lecture Notes in StatisticsSpringer New York NY USA 1998
[13] E Liebscher ldquoEstimation of the density and the regressionfunction under mixing conditionsrdquo Statistics and Decisions vol19 no 1 pp 9ndash26 2001
[14] D L Donoho and I M Johnstone ldquoIdeal spatial adaptation bywavelet shrinkagerdquo Biometrika vol 81 no 3 pp 425ndash455 1994
[15] D L Donoho and I M Johnstone ldquoAdapting to unknownsmoothness via wavelet shrinkagerdquo Journal of the AmericanStatistical Association vol 90 pp 1200ndash1224 1995
[16] D Donoho I Johnstone G Kerkyacharian and D PicardldquoWavelet shrinkage asymptopiardquo Journal of the Royal StatisticalSociety B vol 57 pp 301ndash369 1995
[17] D L Donoho I M Johnstone G Kerkyacharian and DPicard ldquoDensity estimation by wavelet thresholdingrdquo Annals ofStatistics vol 24 no 2 pp 508ndash539 1996
[18] A Antoniadis ldquoWavelets in statistics a review (with discus-sion)rdquo Journal of the Italian Statistical Society vol 6 pp 97ndash1441997
[19] W Hardle G Kerkyacharian D Picard and A TsybakovWavelet Approximation and Statistical Applications vol 129 ofLectures Notes in Statistics Springer New York NY USA 1998
[20] F Leblanc ldquoWavelet linear density estimator for a discrete-timestochastic process 119871
119901-lossesrdquo Statistics and Probability Letters
vol 27 no 1 pp 71ndash84 1996[21] K Tribouley andG Viennet ldquo119871
119901adaptive density estimation in
a alpha-mixing frameworkrdquo Annales de lrsquoinstitut Henri PoincareB vol 34 no 2 pp 179ndash208 1998
[22] E Masry ldquoWavelet-based estimation of multivariate regressionfunctions in besov spacesrdquo Journal of Nonparametric Statisticsvol 12 no 2 pp 283ndash308 2000
[23] P N Patil and Y K Truong ldquoAsymptotics for wavelet basedestimates of piecewise smooth regression for stationary timeseriesrdquo Annals of the Institute of Statistical Mathematics vol 53no 1 pp 159ndash178 2001
[24] H Doosti M Afshari and H A Niroumand ldquoWavelets fornonparametric stochastic regression with mixing stochasticprocessrdquo Communications in Statistics vol 37 no 3 pp 373ndash385 2008
[25] H Doosti and H A Niroumand ldquoMultivariate stochasticregression estimation by wavelet methods for stationary timeseriesrdquo Pakistan Journal of Statistics vol 27 no 1 pp 37ndash462009
[26] H Doosti M S Islam Y P Chaubey and P Gora ldquoTwo-dimensional wavelets for nonlinear autoregressive models withan application in dynamical systemrdquo Italian Journal of Pure andApplied Mathematics no 27 pp 39ndash62 2010
[27] J Cai and H Liang ldquoNonlinear wavelet density estimation fortruncated and dependent observationsrdquo International Journal ofWavelets Multiresolution and Information Processing vol 9 no4 pp 587ndash609 2011
[28] S Niu and H Liang ldquoNonlinear wavelet estimation of condi-tional density under left-truncated and 120572-mixing assumptionsrdquoInternational Journal of Wavelets Multiresolution and Informa-tion Processing vol 9 no 6 pp 989ndash1023 2011
[29] F Benatia and D Yahia ldquoNonlinear wavelet regression functionestimator for censored dependent datardquo Journal Afrika Statis-tika vol 7 no 1 pp 391ndash411 2012
[30] C Chesneau ldquoOn the adaptive wavelet deconvolution of adensity for strong mixing sequencesrdquo Journal of the KoreanStatistical Society vol 41 no 4 pp 423ndash436 2012
[31] C Chesneau ldquoWavelet estimation of a density in a GARCH-type modelrdquo Communications in Statistics vol 42 no 1 pp 98ndash117 2013
[32] C Chesneau ldquoOn the adaptive wavelet estimation of a mul-tidimensional regression function under alpha-mixing depen-dence beyond the standard assumptions on the noiserdquo Com-mentationes Mathematicae Universitatis Carolinae vol 4 pp527ndash556 2013
[33] Y P Chaubey and E Shirazi ldquoOn MISE of a nonlinear waveletestimator of the regression function based on biased data understrong mixing rdquo Communications in Statistics In press
[34] M Abbaszadeh andM Emadi ldquoWavelet density estimation andstatistical evidences role for a garch model in the weighteddistributionrdquo Applied Mathematics vol 4 no 2 pp 10ndash4162013
[35] B Delyon and A Juditsky ldquoOn minimax wavelet estimatorsrdquoApplied and Computational Harmonic Analysis vol 3 no 3 pp215ndash228 1996
[36] G Kerkyacharian and D Picard ldquoThresholding algorithmsmaxisets and well-concentrated basesrdquo Test vol 9 no 2 pp283ndash344 2000
[37] I Daubechies Ten Lectures on Wavelets SIAM 1992[38] A Cohen I Daubechies and P Vial ldquoWavelets on the interval
and fast wavelet transformsrdquo Applied and Computational Har-monic Analysis vol 1 no 1 pp 54ndash81 1993
[39] S Mallat A Wavelet Tour of Signal Processing The sparse waywith contributions fromGabriel Peyralphae ElsevierAcademicPress Amsterdam The Netherlands 3rd edition 2009
12 Journal of Probability and Statistics
[40] Y Meyer Ondelettes et Opalphaerateurs Hermann ParisFrance 1990
[41] V Genon-Catalot T Jeantheau and C Laredo ldquoStochasticvolatility models as hidden Markov models and statisticalapplicationsrdquo Bernoulli vol 6 no 6 pp 1051ndash1079 2000
[42] J Fan and J Koo ldquoWavelet deconvolutionrdquo IEEE Transactionson Information Theory vol 48 no 3 pp 734ndash747 2002
[43] A B Tsybakov Introduction alphaa lrsquoestimation non-paramalphaetrique Springer New York NY USA 2004
[44] E Masry ldquoStrong consistency and rates for deconvolutionof multivariate densities of stationary processesrdquo StochasticProcesses and their Applications vol 47 no 1 pp 53ndash74 1993
[45] R Kulik ldquoNonparametric deconvolution problem for depen-dent sequencesrdquo Electronic Journal of Statistics vol 2 pp 722ndash740 2008
[46] F Comte J Dedecker and M L Taupin ldquoAdaptive densitydeconvolution with dependent inputsrdquo Mathematical Methodsof Statistics vol 17 no 2 pp 87ndash112 2008
[47] H van Zanten and P Zareba ldquoA note on wavelet densitydeconvolution for weakly dependent datardquo Statistical Inferencefor Stochastic Processes vol 11 no 2 pp 207ndash219 2008
[48] W Hardle Applied Nonparametric Regression Cambridge Uni-versity Press Cambridge UK 1990
[49] V A Vasiliev ldquoOne investigation method of a ratios typeestimatorsrdquo in Proceedings of the 16th IFAC Symposium onSystem Identialphacation pp 1ndash6 Brussels Belgium July 2012(in progress)
[50] Y Davydov ldquoThe invariance principle for stationary processesrdquoTheory of Probability and Its Applications vol 15 no 3 pp 498ndash509 1970
[51] C Chesneau ldquoWavelet estimation via block thresholding aminimax study under 119871119901-riskrdquo Statistica Sinica vol 18 no 3pp 1007ndash1024 2008
[52] C Chesneau and H Doosti ldquoWavelet linear density estimationfor a GARCH model under various dependence structuresrdquoJournal of the Iranian Statistical Society vol 11 no 1 pp 1ndash212012
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
8 Journal of Probability and Statistics
Let us now investigate sharp upper bounds for 119875 119877 and119876 successively
Upper Bound for 119875 The point (a) of Proposition 10 and2119904(2119904 + 2120588 + 1) lt 1 yield
119875 le 11986221198950
119899le 119862
ln 119899
119899le 119862(
ln 119899
119899)
2119904(2119904+2120588+1)
(66)
Upper Bound for 119877(i) For 119903 ge 1 and119901 ge 2 we have119891 isin 119861119904
119901119903(119872) sube 119861119904
2infin(119872)
Using 2119904(2119904 + 2120588 + 1) lt 2119904(2120588 + 1) we obtain
119877 le 119862infin
sum119895=1198951+1
2minus2119895119904 le 119862((ln 119899)3
119899)
2119904(2120588+1)
le 119862(ln 119899
119899)
2119904(2119904+2120588+1)
(67)
(ii) For 119903 ge 1 and 119901 isin [1 2) we have 119891 isin 119861119904
119901119903(119872) sube
119861119904+12minus1119901
2infin(119872) The condition 119904 gt (2120588 + 1)119901 implies
that (119904 + 12 minus 1119901)(2120588 + 1) gt 119904(2119904 + 2120588 + 1) Thus
119877 le 119862infin
sum119895=1198951+1
2minus2119895(119904+12minus1119901)
le 119862((ln 119899)3
119899)
2(119904+12minus1119901)(2120588+1)
le 119862(ln 119899
119899)
2119904(2119904+2120588+1)
(68)
Hence for 119903 ge 1 119901 ge 2 and 119904 gt 0 or 119901 isin [1 2) and119904 gt (2120588 + 1)119901 we have
119877 le 119862(ln 119899
119899)
2119904(2119904+2120588+1)
(69)
Upper Bound for 119876 Adopting the notation119863119895119896
= 119889119895119896
minus 119889119895119896
119876 can be written as
119876 =4
sum119894=1
119876119894 (70)
where
1198761=
1198951
sum119895=1198950
sum119896isinΛ 119895
E (10038161003816100381610038161003816119863119895119896
100381610038161003816100381610038162
1|119889119895119896|ge120581120582119895 |119889119895119896|lt1205811205821198952
)
1198762=
1198951
sum119895=1198950
sum119896isinΛ 119895
E (10038161003816100381610038161003816119863119895119896
100381610038161003816100381610038162
1|119889119895119896|ge120581120582119895 |119889119895119896|ge1205811205821198952
)
1198763=
1198951
sum119895=1198950
sum119896isinΛ 119895
E (1198892
1198951198961|119889119895119896|lt120581120582119895 |119889119895119896|ge2120581120582119895
)
1198764=
1198951
sum119895=1198950
sum119896isinΛ 119895
E (1198892
1198951198961|119889119895119896|lt120581120582119895 |119889119895119896|lt2120581120582119895
)
(71)
Upper Bound for 1198761
+ 1198763 Owing to the inequalities
1|119889119895119896|lt120581120582119895 |119889119895119896|ge2120581120582119895
le 1|119895119896|gt1205811205821198952
1|119889119895119896|ge120581120582119895 |119889119895119896|lt1205811205821198952
le
1|119895119896|gt1205811205821198952
and 1|119889119895119896|lt120581120582119895 |119889119895119896|ge2120581120582119895
le 1|119889119895119896|le2|119895119896|
theCauchy-Schwarz inequality and the points(b) and (c) ofProposition 10 we have
1198761+ 119876
3le 119862
1198951
sum119895=1198950
sum119896isinΛ 119895
E (10038161003816100381610038161003816119863119895119896
100381610038161003816100381610038162
1|119895119896|gt1205811205821198952
)
le 119862
1198951
sum119895=1198950
sum119896isinΛ 119895
(E (10038161003816100381610038161003816119863119895119896
100381610038161003816100381610038164
))12
(P(10038161003816100381610038161003816119863119895119896
10038161003816100381610038161003816 gt120581120582
119895
2))
12
le 1198621
1198992
1198951
sum119895=1198950
2119895(1+2120588) le 1198621
119899le 119862(
ln 119899
119899)
2119904(2119904+2120588+1)
(72)
Upper Bound for 1198762 It follows from the point(a) of
Proposition 10 that
1198762le
1198951
sum119895=1198950
sum119896isinΛ 119895
E (10038161003816100381610038161003816119863119895119896
100381610038161003816100381610038162
) 1|119889119895119896|ge1205811205821198952
le 1198621
119899
1198951
sum119895=j0
22120588119895 sum119896isinΛ 119895
1|119889119895119896|gt1205811205821198952
(73)
Let us now introduce the integer 119895lowastdefined by
2119895lowast = [(119899
ln 119899)
1(2119904+2120588+1)
] (74)
Note that 119895lowastisin 119895
0 119895
1 for 119899 large enough
Then 1198762can be bounded as
1198762le 119876
21+ 119876
22 (75)
where
11987621
= 1198621
119899
119895lowast
sum119895=1198950
22120588119895 sum119896isinΛ 119895
1|119889119895119896|gt1205811205821198952
11987622
= 1198621
119899
1198951
sum119895=119895lowast+1
22120588119895 sum119896isinΛ 119895
1|119889119895119896|gt1205811205821198952
(76)
On the one hand we have
11987621
le 119862ln 119899
119899
119895lowast
sum119895=1198950
2119895(1+2120588) le 119862(ln 119899
119899)
2119904(2119904+2120588+1)
(77)
On the other hand we have the following
Journal of Probability and Statistics 9
(i) For 119903 ge 1 and 119901 ge 2 the Markov inequality and 119891 isin119861119904
119901119903(119872) sube 119861119904
2infin(119872) yield
11987622
le 119862ln 119899
119899
1198951
sum119895=119895lowast+1
22120588119895 1
1205822
119895
sum119896isinΛ 119895
1198892
119895119896le 119862
infin
sum119895=119895lowast+1
sum119896isinΛ 119895
1198892
119895119896
le 119862infin
sum119895=119895lowast+1
2minus2119895119904 le 119862(ln 119899
119899)
2119904(2119904+2120588+1)
(78)
(ii) For 119903 ge 1 119901 isin [1 2) and 119904 gt (2120588 + 1)119901 the Markovinequality 119891 isin 119861119904
119901119903(119872) and (2119904 + 2120588 + 1)(2 minus 119901)2 +
(119904 + 12 minus 1119901 + 120588 minus 2120588119901)119901 = 2119904 imply that
11987622
le 119862ln 119899
119899
1198951
sum119895=119895lowast+1
22120588119895 1
120582119901
119895
sum119896isinΛ 119895
10038161003816100381610038161003816119889119895119896
10038161003816100381610038161003816119901
le 119862(ln 119899
119899)
(2minus119901)2 infin
sum119895=119895lowast+1
2119895120588(2minus119901)2minus119895(119904+12minus1119901)119901
le 119862(ln 119899
119899)
(2minus119901)2
2minus119895lowast(119904+12minus1119901+120588minus2120588119901)119901
le 119862(ln 119899
119899)
2119904(2119904+2120588+1)
(79)
Therefore for 119903 ge 1 119901 ge 2 and 119904 gt 0 or 119901 isin [1 2) and119904 gt (2120588 + 1)119901 we have
1198762le C(
ln 119899
119899)
2119904(2119904+2120588+1)
(80)
Upper Bound for 1198764 We have
1198764le
1198951
sum119895=1198950
sum119896isinΛ 119895
1198892
1198951198961|119889119895119896|lt2120581120582119895
(81)
Let 119895lowastbe the integer (74) Then 119876
4can be bound as
1198764le 119876
41+ 119876
42 (82)
where
11987641
=
119895lowast
sum119895=1198950
sum119896isinΛ 119895
1198892
1198951198961|119889119895119896|lt2120581120582119895
11987642
=
1198951
sum119895=119895lowast+1
sum119896isinΛ 119895
1198892
1198951198961|119889119895119896|lt2120581120582119895
(83)
On the one hand we have
11987641
le 119862
119895lowast
sum119895=1198950
21198951205822
119895= 119862
ln 119899
119899
119895lowast
sum119895=1198950
2119895(1+2120588) le 119862(ln 119899
119899)
2119904(2119904+2120588+1)
(84)
On the other hand we have the following
(i) For 119903 ge 1 and 119901 ge 2 since 119891 isin 119861119904
119901119903(119872) sube 119861119904
2infin(119872)
we have
11987642
leinfin
sum119895=119895lowast+1
sum119896isinΛ 119895
1198892
119895119896le 119862
infin
sum119895=119895lowast+1
2minus2119895119904 le 119862(ln 119899
119899)
2119904(2119904+2120588+1)
(85)
(ii) For 119903 ge 1 119901 isin [1 2) and 119904 gt (2120588 + 1)119901 owing to theMarkov inequality 119891 isin 119861119904
119901119903(119872) and (2119904 + 2120588 + 1)(2 minus
119901)2 + (119904 + 12 minus 1119901 + 120588 minus 2120588119901)119901 = 2119904 we get
11987642
le 119862
1198951
sum119895=119895lowast+1
1205822minus119901
119895sum
119896isinΛ 119895
10038161003816100381610038161003816119889119895119896
10038161003816100381610038161003816119901
= 119862(ln 119899
119899)
(2minus119901)2 1198951
sum119895=119895lowast+1
2119895120588(2minus119901) sum119896isinΛ 119895
10038161003816100381610038161003816119889119895119896
10038161003816100381610038161003816119901
le 119862(ln 119899
119899)
(2minus119901)2 infin
sum119895=119895lowast+1
2119895120588(2minus119901)2minus119895(119904+12minus1119901)119901
le 119862(ln 119899
119899)
(2minus119901)2
2minus119895lowast(119904+12minus1119901+120588minus2120588119901)119901
le 119862(ln 119899
119899)
2119904(2119904+2120588+1)
(86)
So for 119903 ge 1 119901 ge 2 and 119904 gt 0 or 119901 isin [1 2) and 119904 gt(2120588 + 1)119901 we have
1198764le 119862(
ln 119899
119899)
2119904(2119904+2120588+1)
(87)
Putting (70) (72) (80) and (87) together for 119903 ge 1 119901 ge 2and 119904 gt 0 or 119901 isin [1 2) and 119904 gt (2120588 + 1)119901 we obtain
119876 le 119862(ln 119899
119899)
2119904(2119904+2120588+1)
(88)
Combining (64) (66) (69) and (88) we complete theproof of Theorem 3
Proof of Theorem 4 The proof of Theorem 4 is a direct appli-cation ofTheorem 3 under (G1)ndash(G5) the function 119902 definedby (24) satisfies (H1) see ([42] equation (2)) and (H2) (i) see([42] Lemma 6) (ii) see ([42] equation (11)) and (iii) see([30] Proof of Proposition 61) with 120588 = 120575
Proof of Theorem 5 The proof ofTheorem 5 is a consequenceofTheorem 3 under (J1)ndash(J4) the function 119902 defined by (30)satisfies (H1) and (H2) (i)-(ii) see ([31] Proposition 1) and(iii) see ([52] equation (26)) with 120588 = 120575
Proof of Theorem 6 Set V(119909) = 119891(119909)119892(119909) Following themethodology of [49] we have
119891 (119909) minus 119891 (119909) = 119878 (119909) minus 119879 (119909) (89)
10 Journal of Probability and Statistics
where
119878 (119909)
=1
119892 (119909)(V (119909) minus V (119909)
+119891 (119909) (119892 (119909) minus 119892 (119909))) 1|119892(119909)|ge119888lowast2
119879 (119909) = 119891 (119909) 1|119892(119909)|lt119888lowast2
(90)
Using (K3) and the indicator function we have
|119878 (119909)| le 119862 (|V (119909) minus V (119909)| + 1003816100381610038161003816119892 (119909) minus 119892 (119909)1003816100381610038161003816) (91)
It follows from |119892(119909)| lt 119888lowast2 cap |119892(119909)| gt 119888
lowast sube |119892(119909) minus
119892(119909)| gt 119888lowast2 (K3) (K4) and the Markov inequality that
|119879 (119909)| le 1198621|119892(119909)minus119892(119909)|gt119888lowast2
le 1198621003816100381610038161003816119892 (119909) minus 119892 (119909)
1003816100381610038161003816 (92)
The triangular inequality yields
10038161003816100381610038161003816119891 (119909) minus 119891 (119909)10038161003816100381610038161003816 le 119862 (|V (119909) minus V (119909)| + 1003816100381610038161003816119892 (119909) minus 119892 (119909)
1003816100381610038161003816) (93)
The elementary inequality (119886 + 119887)2 le 2(1198862 + 1198872) implies that
E (10038171003817100381710038171003817119891 minus 119891
100381710038171003817100381710038172
2) le 119862 (E (V minus V2
2) + E (
1003817100381710038171003817119892 minus 11989210038171003817100381710038172
2)) (94)
We now bound this two MISEs via Theorem 3
Upper Bound for the MISE of V Under (K1)ndash(K6) thefunction 119902 defined by (38) satisfies the following
(H1) With V instead of 119891 since 1205851and 119883
1are independent
with E(1205851) = 0
E (119902 (120574119895119896
1198811))
= E (1198841120574119895119896
(1198831)) = E (119891 (119883
1) 120574
119895119896(119883
1))
= int1
0
119891 (119909) 120574119895119896
(119909) 119892 (119909) 119889119909 = int1
0
V (119909) 120574119895119896
(119909) 119889119909
(95)
(H2) (i)-(ii)-(iii) with 120588 = 0
(i) since 1198841(Ω) is bounded thanks to (K2) and (K3) say
|1198841| le 119872 with 119872 gt 0 we have
sup(119909119909lowast)isin1198811(Ω)
10038161003816100381610038161003816119902 (120574119895119896
(119909 119909lowast))
10038161003816100381610038161003816
= sup(119909119909lowast)isin[minus119872119872]times[01]
10038161003816100381610038161003816119909120574119895119896(119909
lowast)10038161003816100381610038161003816
le 119872 sup119909lowastisin[01]
10038161003816100381610038161003816120574119895119896(119909
lowast)10038161003816100381610038161003816 le 11986221198952
(96)
(ii) using the boundedness of 1198841(Ω) then (K4) we have
E (10038161003816100381610038161003816119902(120574119895119896
1198811)100381610038161003816100381610038162
) = E (1198842
1(120574
119895119896(119883
1))
2
)
le 119862E ((120574119895119896
(1198831))
2
)
= 119862int1
0
(120574119895119896
(119909))2
119892 (119909) 119889119909
le 119862int1
0
(120574119895119896
(119909))2
119889119909 le 119862
(97)
(iii) using the boundedness of 1198841(Ω) and making the
change of variables 119910 = 2119895119909 minus 119896 we obtain
int1198811(Ω)
10038161003816100381610038161003816119902 (120574119895119896
119909)10038161003816100381610038161003816 119889119909
= (int119872
minus119872
|119909| 119889119909)(int1
0
10038161003816100381610038161003816120574119895119896(119909
lowast)10038161003816100381610038161003816 119889119909lowast
)
= 1198722 int1
0
10038161003816100381610038161003816120574119895119896(119909)
10038161003816100381610038161003816 119889119909 le 1198622minus1198952
(98)
We conclude by applying Lemma 2 with 120588 = 0 (K5) and (K6)imply (F1) and the previous inequality implies (F2)
Therefore assuming that V isin 119861119904
119901119903(119872) with 119903 ge 1 119901 ge 2
and 119904 isin (0119873) or 119901 isin [1 2) and 119904 isin (1119901119873) Theorem 3proves the existence of a constant 119862 gt 0 satisfying
E (V minus V22) le 119862(
ln 119899
119899)
2119904(2119904+1)
(99)
Upper Bound for theMISE of 119892 Under (K1)ndash(K6) proceedingas the previous point we show that the function 119902 defined by(39) satisfies (H1) with 119892 instead of 119891 and 119883
119905instead of 119881
119905
and(H2) (i)-(ii)-(iii) with 120588 = 0Therefore assuming that 119892 isin 119861119904
119901119903(119872) with 119903 ge 1 119901 ge 2
and 119904 isin (0119873) or 119901 isin [1 2) and 119904 isin (1119901119873) Theorem 3proves the existance of a constant 119862 gt 0 satisfying
E (1003817100381710038171003817119892 minus 119892
10038171003817100381710038172
2) le 119862(
ln 119899
119899)
2119904(2119904+1)
(100)
Combining (94) (99) and (100) we end the proof ofTheorem 6
Conflict of Interests
The author declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
The author is thankful to the reviewers for their commentswhich have helped in improving the presentation
Journal of Probability and Statistics 11
References
[1] P DoukhanMixing Properties and Examples vol 85 of LectureNotes in Statistics Springer New York NY USA 1994
[2] M Carrasco and X Chen ldquoMixing and moment properties ofvarious GARCH and stochastic volatility modelsrdquo EconometricTheory vol 18 no 1 pp 17ndash39 2002
[3] R C Bradley Introduction to Strong Mixing Conditions Volume1 2 3 Kendrick Press 2007
[4] P M Robinson ldquoNonparametric estimators for time seriesrdquoJournal of Time Series Analysis vol 4 pp 185ndash207 1983
[5] G G Roussas ldquoNonparametric estimation inmixing sequencesof random variablesrdquo Journal of Statistical Planning and Infer-ence vol 18 no 2 pp 135ndash149 1987
[6] G G Roussas ldquoNonparametric regression estimation undermixing conditionsrdquo Stochastic Processes and their Applicationsvol 36 no 1 pp 107ndash116 1990
[7] Y K Truong and C J Stone ldquoNonparametric function estima-tion involving time seriesrdquo The Annals of Statistics vol 20 pp77ndash97 1992
[8] L H Tran ldquoNonparametric function estimation for time seriesby local average estimatorsrdquoThe Annals of Statistics vol 21 pp1040ndash1057 1993
[9] E Masry ldquoMultivariate local polynomial regression for timeseries uniform strong consistency and ratesrdquo Journal of TimeSeries Analysis vol 17 no 6 pp 571ndash599 1996
[10] E Masry ldquoMultivariate regression estimation local polynomialfitting for time seriesrdquo Stochastic Processes and their Applica-tions vol 65 no 1 pp 81ndash101 1996
[11] E Masry and J Fan ldquoLocal polynomial estimation of regressionfunctions for mixing processesrdquo Scandinavian Journal of Statis-tics vol 24 no 2 pp 165ndash179 1997
[12] D Bosq Nonparametric Statistics for Stochastic Processes Esti-mation and Prediction vol 110 of Lecture Notes in StatisticsSpringer New York NY USA 1998
[13] E Liebscher ldquoEstimation of the density and the regressionfunction under mixing conditionsrdquo Statistics and Decisions vol19 no 1 pp 9ndash26 2001
[14] D L Donoho and I M Johnstone ldquoIdeal spatial adaptation bywavelet shrinkagerdquo Biometrika vol 81 no 3 pp 425ndash455 1994
[15] D L Donoho and I M Johnstone ldquoAdapting to unknownsmoothness via wavelet shrinkagerdquo Journal of the AmericanStatistical Association vol 90 pp 1200ndash1224 1995
[16] D Donoho I Johnstone G Kerkyacharian and D PicardldquoWavelet shrinkage asymptopiardquo Journal of the Royal StatisticalSociety B vol 57 pp 301ndash369 1995
[17] D L Donoho I M Johnstone G Kerkyacharian and DPicard ldquoDensity estimation by wavelet thresholdingrdquo Annals ofStatistics vol 24 no 2 pp 508ndash539 1996
[18] A Antoniadis ldquoWavelets in statistics a review (with discus-sion)rdquo Journal of the Italian Statistical Society vol 6 pp 97ndash1441997
[19] W Hardle G Kerkyacharian D Picard and A TsybakovWavelet Approximation and Statistical Applications vol 129 ofLectures Notes in Statistics Springer New York NY USA 1998
[20] F Leblanc ldquoWavelet linear density estimator for a discrete-timestochastic process 119871
119901-lossesrdquo Statistics and Probability Letters
vol 27 no 1 pp 71ndash84 1996[21] K Tribouley andG Viennet ldquo119871
119901adaptive density estimation in
a alpha-mixing frameworkrdquo Annales de lrsquoinstitut Henri PoincareB vol 34 no 2 pp 179ndash208 1998
[22] E Masry ldquoWavelet-based estimation of multivariate regressionfunctions in besov spacesrdquo Journal of Nonparametric Statisticsvol 12 no 2 pp 283ndash308 2000
[23] P N Patil and Y K Truong ldquoAsymptotics for wavelet basedestimates of piecewise smooth regression for stationary timeseriesrdquo Annals of the Institute of Statistical Mathematics vol 53no 1 pp 159ndash178 2001
[24] H Doosti M Afshari and H A Niroumand ldquoWavelets fornonparametric stochastic regression with mixing stochasticprocessrdquo Communications in Statistics vol 37 no 3 pp 373ndash385 2008
[25] H Doosti and H A Niroumand ldquoMultivariate stochasticregression estimation by wavelet methods for stationary timeseriesrdquo Pakistan Journal of Statistics vol 27 no 1 pp 37ndash462009
[26] H Doosti M S Islam Y P Chaubey and P Gora ldquoTwo-dimensional wavelets for nonlinear autoregressive models withan application in dynamical systemrdquo Italian Journal of Pure andApplied Mathematics no 27 pp 39ndash62 2010
[27] J Cai and H Liang ldquoNonlinear wavelet density estimation fortruncated and dependent observationsrdquo International Journal ofWavelets Multiresolution and Information Processing vol 9 no4 pp 587ndash609 2011
[28] S Niu and H Liang ldquoNonlinear wavelet estimation of condi-tional density under left-truncated and 120572-mixing assumptionsrdquoInternational Journal of Wavelets Multiresolution and Informa-tion Processing vol 9 no 6 pp 989ndash1023 2011
[29] F Benatia and D Yahia ldquoNonlinear wavelet regression functionestimator for censored dependent datardquo Journal Afrika Statis-tika vol 7 no 1 pp 391ndash411 2012
[30] C Chesneau ldquoOn the adaptive wavelet deconvolution of adensity for strong mixing sequencesrdquo Journal of the KoreanStatistical Society vol 41 no 4 pp 423ndash436 2012
[31] C Chesneau ldquoWavelet estimation of a density in a GARCH-type modelrdquo Communications in Statistics vol 42 no 1 pp 98ndash117 2013
[32] C Chesneau ldquoOn the adaptive wavelet estimation of a mul-tidimensional regression function under alpha-mixing depen-dence beyond the standard assumptions on the noiserdquo Com-mentationes Mathematicae Universitatis Carolinae vol 4 pp527ndash556 2013
[33] Y P Chaubey and E Shirazi ldquoOn MISE of a nonlinear waveletestimator of the regression function based on biased data understrong mixing rdquo Communications in Statistics In press
[34] M Abbaszadeh andM Emadi ldquoWavelet density estimation andstatistical evidences role for a garch model in the weighteddistributionrdquo Applied Mathematics vol 4 no 2 pp 10ndash4162013
[35] B Delyon and A Juditsky ldquoOn minimax wavelet estimatorsrdquoApplied and Computational Harmonic Analysis vol 3 no 3 pp215ndash228 1996
[36] G Kerkyacharian and D Picard ldquoThresholding algorithmsmaxisets and well-concentrated basesrdquo Test vol 9 no 2 pp283ndash344 2000
[37] I Daubechies Ten Lectures on Wavelets SIAM 1992[38] A Cohen I Daubechies and P Vial ldquoWavelets on the interval
and fast wavelet transformsrdquo Applied and Computational Har-monic Analysis vol 1 no 1 pp 54ndash81 1993
[39] S Mallat A Wavelet Tour of Signal Processing The sparse waywith contributions fromGabriel Peyralphae ElsevierAcademicPress Amsterdam The Netherlands 3rd edition 2009
12 Journal of Probability and Statistics
[40] Y Meyer Ondelettes et Opalphaerateurs Hermann ParisFrance 1990
[41] V Genon-Catalot T Jeantheau and C Laredo ldquoStochasticvolatility models as hidden Markov models and statisticalapplicationsrdquo Bernoulli vol 6 no 6 pp 1051ndash1079 2000
[42] J Fan and J Koo ldquoWavelet deconvolutionrdquo IEEE Transactionson Information Theory vol 48 no 3 pp 734ndash747 2002
[43] A B Tsybakov Introduction alphaa lrsquoestimation non-paramalphaetrique Springer New York NY USA 2004
[44] E Masry ldquoStrong consistency and rates for deconvolutionof multivariate densities of stationary processesrdquo StochasticProcesses and their Applications vol 47 no 1 pp 53ndash74 1993
[45] R Kulik ldquoNonparametric deconvolution problem for depen-dent sequencesrdquo Electronic Journal of Statistics vol 2 pp 722ndash740 2008
[46] F Comte J Dedecker and M L Taupin ldquoAdaptive densitydeconvolution with dependent inputsrdquo Mathematical Methodsof Statistics vol 17 no 2 pp 87ndash112 2008
[47] H van Zanten and P Zareba ldquoA note on wavelet densitydeconvolution for weakly dependent datardquo Statistical Inferencefor Stochastic Processes vol 11 no 2 pp 207ndash219 2008
[48] W Hardle Applied Nonparametric Regression Cambridge Uni-versity Press Cambridge UK 1990
[49] V A Vasiliev ldquoOne investigation method of a ratios typeestimatorsrdquo in Proceedings of the 16th IFAC Symposium onSystem Identialphacation pp 1ndash6 Brussels Belgium July 2012(in progress)
[50] Y Davydov ldquoThe invariance principle for stationary processesrdquoTheory of Probability and Its Applications vol 15 no 3 pp 498ndash509 1970
[51] C Chesneau ldquoWavelet estimation via block thresholding aminimax study under 119871119901-riskrdquo Statistica Sinica vol 18 no 3pp 1007ndash1024 2008
[52] C Chesneau and H Doosti ldquoWavelet linear density estimationfor a GARCH model under various dependence structuresrdquoJournal of the Iranian Statistical Society vol 11 no 1 pp 1ndash212012
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Journal of Probability and Statistics 9
(i) For 119903 ge 1 and 119901 ge 2 the Markov inequality and 119891 isin119861119904
119901119903(119872) sube 119861119904
2infin(119872) yield
11987622
le 119862ln 119899
119899
1198951
sum119895=119895lowast+1
22120588119895 1
1205822
119895
sum119896isinΛ 119895
1198892
119895119896le 119862
infin
sum119895=119895lowast+1
sum119896isinΛ 119895
1198892
119895119896
le 119862infin
sum119895=119895lowast+1
2minus2119895119904 le 119862(ln 119899
119899)
2119904(2119904+2120588+1)
(78)
(ii) For 119903 ge 1 119901 isin [1 2) and 119904 gt (2120588 + 1)119901 the Markovinequality 119891 isin 119861119904
119901119903(119872) and (2119904 + 2120588 + 1)(2 minus 119901)2 +
(119904 + 12 minus 1119901 + 120588 minus 2120588119901)119901 = 2119904 imply that
11987622
le 119862ln 119899
119899
1198951
sum119895=119895lowast+1
22120588119895 1
120582119901
119895
sum119896isinΛ 119895
10038161003816100381610038161003816119889119895119896
10038161003816100381610038161003816119901
le 119862(ln 119899
119899)
(2minus119901)2 infin
sum119895=119895lowast+1
2119895120588(2minus119901)2minus119895(119904+12minus1119901)119901
le 119862(ln 119899
119899)
(2minus119901)2
2minus119895lowast(119904+12minus1119901+120588minus2120588119901)119901
le 119862(ln 119899
119899)
2119904(2119904+2120588+1)
(79)
Therefore for 119903 ge 1 119901 ge 2 and 119904 gt 0 or 119901 isin [1 2) and119904 gt (2120588 + 1)119901 we have
1198762le C(
ln 119899
119899)
2119904(2119904+2120588+1)
(80)
Upper Bound for 1198764 We have
1198764le
1198951
sum119895=1198950
sum119896isinΛ 119895
1198892
1198951198961|119889119895119896|lt2120581120582119895
(81)
Let 119895lowastbe the integer (74) Then 119876
4can be bound as
1198764le 119876
41+ 119876
42 (82)
where
11987641
=
119895lowast
sum119895=1198950
sum119896isinΛ 119895
1198892
1198951198961|119889119895119896|lt2120581120582119895
11987642
=
1198951
sum119895=119895lowast+1
sum119896isinΛ 119895
1198892
1198951198961|119889119895119896|lt2120581120582119895
(83)
On the one hand we have
11987641
le 119862
119895lowast
sum119895=1198950
21198951205822
119895= 119862
ln 119899
119899
119895lowast
sum119895=1198950
2119895(1+2120588) le 119862(ln 119899
119899)
2119904(2119904+2120588+1)
(84)
On the other hand we have the following
(i) For 119903 ge 1 and 119901 ge 2 since 119891 isin 119861119904
119901119903(119872) sube 119861119904
2infin(119872)
we have
11987642
leinfin
sum119895=119895lowast+1
sum119896isinΛ 119895
1198892
119895119896le 119862
infin
sum119895=119895lowast+1
2minus2119895119904 le 119862(ln 119899
119899)
2119904(2119904+2120588+1)
(85)
(ii) For 119903 ge 1 119901 isin [1 2) and 119904 gt (2120588 + 1)119901 owing to theMarkov inequality 119891 isin 119861119904
119901119903(119872) and (2119904 + 2120588 + 1)(2 minus
119901)2 + (119904 + 12 minus 1119901 + 120588 minus 2120588119901)119901 = 2119904 we get
11987642
le 119862
1198951
sum119895=119895lowast+1
1205822minus119901
119895sum
119896isinΛ 119895
10038161003816100381610038161003816119889119895119896
10038161003816100381610038161003816119901
= 119862(ln 119899
119899)
(2minus119901)2 1198951
sum119895=119895lowast+1
2119895120588(2minus119901) sum119896isinΛ 119895
10038161003816100381610038161003816119889119895119896
10038161003816100381610038161003816119901
le 119862(ln 119899
119899)
(2minus119901)2 infin
sum119895=119895lowast+1
2119895120588(2minus119901)2minus119895(119904+12minus1119901)119901
le 119862(ln 119899
119899)
(2minus119901)2
2minus119895lowast(119904+12minus1119901+120588minus2120588119901)119901
le 119862(ln 119899
119899)
2119904(2119904+2120588+1)
(86)
So for 119903 ge 1 119901 ge 2 and 119904 gt 0 or 119901 isin [1 2) and 119904 gt(2120588 + 1)119901 we have
1198764le 119862(
ln 119899
119899)
2119904(2119904+2120588+1)
(87)
Putting (70) (72) (80) and (87) together for 119903 ge 1 119901 ge 2and 119904 gt 0 or 119901 isin [1 2) and 119904 gt (2120588 + 1)119901 we obtain
119876 le 119862(ln 119899
119899)
2119904(2119904+2120588+1)
(88)
Combining (64) (66) (69) and (88) we complete theproof of Theorem 3
Proof of Theorem 4 The proof of Theorem 4 is a direct appli-cation ofTheorem 3 under (G1)ndash(G5) the function 119902 definedby (24) satisfies (H1) see ([42] equation (2)) and (H2) (i) see([42] Lemma 6) (ii) see ([42] equation (11)) and (iii) see([30] Proof of Proposition 61) with 120588 = 120575
Proof of Theorem 5 The proof ofTheorem 5 is a consequenceofTheorem 3 under (J1)ndash(J4) the function 119902 defined by (30)satisfies (H1) and (H2) (i)-(ii) see ([31] Proposition 1) and(iii) see ([52] equation (26)) with 120588 = 120575
Proof of Theorem 6 Set V(119909) = 119891(119909)119892(119909) Following themethodology of [49] we have
119891 (119909) minus 119891 (119909) = 119878 (119909) minus 119879 (119909) (89)
10 Journal of Probability and Statistics
where
119878 (119909)
=1
119892 (119909)(V (119909) minus V (119909)
+119891 (119909) (119892 (119909) minus 119892 (119909))) 1|119892(119909)|ge119888lowast2
119879 (119909) = 119891 (119909) 1|119892(119909)|lt119888lowast2
(90)
Using (K3) and the indicator function we have
|119878 (119909)| le 119862 (|V (119909) minus V (119909)| + 1003816100381610038161003816119892 (119909) minus 119892 (119909)1003816100381610038161003816) (91)
It follows from |119892(119909)| lt 119888lowast2 cap |119892(119909)| gt 119888
lowast sube |119892(119909) minus
119892(119909)| gt 119888lowast2 (K3) (K4) and the Markov inequality that
|119879 (119909)| le 1198621|119892(119909)minus119892(119909)|gt119888lowast2
le 1198621003816100381610038161003816119892 (119909) minus 119892 (119909)
1003816100381610038161003816 (92)
The triangular inequality yields
10038161003816100381610038161003816119891 (119909) minus 119891 (119909)10038161003816100381610038161003816 le 119862 (|V (119909) minus V (119909)| + 1003816100381610038161003816119892 (119909) minus 119892 (119909)
1003816100381610038161003816) (93)
The elementary inequality (119886 + 119887)2 le 2(1198862 + 1198872) implies that
E (10038171003817100381710038171003817119891 minus 119891
100381710038171003817100381710038172
2) le 119862 (E (V minus V2
2) + E (
1003817100381710038171003817119892 minus 11989210038171003817100381710038172
2)) (94)
We now bound this two MISEs via Theorem 3
Upper Bound for the MISE of V Under (K1)ndash(K6) thefunction 119902 defined by (38) satisfies the following
(H1) With V instead of 119891 since 1205851and 119883
1are independent
with E(1205851) = 0
E (119902 (120574119895119896
1198811))
= E (1198841120574119895119896
(1198831)) = E (119891 (119883
1) 120574
119895119896(119883
1))
= int1
0
119891 (119909) 120574119895119896
(119909) 119892 (119909) 119889119909 = int1
0
V (119909) 120574119895119896
(119909) 119889119909
(95)
(H2) (i)-(ii)-(iii) with 120588 = 0
(i) since 1198841(Ω) is bounded thanks to (K2) and (K3) say
|1198841| le 119872 with 119872 gt 0 we have
sup(119909119909lowast)isin1198811(Ω)
10038161003816100381610038161003816119902 (120574119895119896
(119909 119909lowast))
10038161003816100381610038161003816
= sup(119909119909lowast)isin[minus119872119872]times[01]
10038161003816100381610038161003816119909120574119895119896(119909
lowast)10038161003816100381610038161003816
le 119872 sup119909lowastisin[01]
10038161003816100381610038161003816120574119895119896(119909
lowast)10038161003816100381610038161003816 le 11986221198952
(96)
(ii) using the boundedness of 1198841(Ω) then (K4) we have
E (10038161003816100381610038161003816119902(120574119895119896
1198811)100381610038161003816100381610038162
) = E (1198842
1(120574
119895119896(119883
1))
2
)
le 119862E ((120574119895119896
(1198831))
2
)
= 119862int1
0
(120574119895119896
(119909))2
119892 (119909) 119889119909
le 119862int1
0
(120574119895119896
(119909))2
119889119909 le 119862
(97)
(iii) using the boundedness of 1198841(Ω) and making the
change of variables 119910 = 2119895119909 minus 119896 we obtain
int1198811(Ω)
10038161003816100381610038161003816119902 (120574119895119896
119909)10038161003816100381610038161003816 119889119909
= (int119872
minus119872
|119909| 119889119909)(int1
0
10038161003816100381610038161003816120574119895119896(119909
lowast)10038161003816100381610038161003816 119889119909lowast
)
= 1198722 int1
0
10038161003816100381610038161003816120574119895119896(119909)
10038161003816100381610038161003816 119889119909 le 1198622minus1198952
(98)
We conclude by applying Lemma 2 with 120588 = 0 (K5) and (K6)imply (F1) and the previous inequality implies (F2)
Therefore assuming that V isin 119861119904
119901119903(119872) with 119903 ge 1 119901 ge 2
and 119904 isin (0119873) or 119901 isin [1 2) and 119904 isin (1119901119873) Theorem 3proves the existence of a constant 119862 gt 0 satisfying
E (V minus V22) le 119862(
ln 119899
119899)
2119904(2119904+1)
(99)
Upper Bound for theMISE of 119892 Under (K1)ndash(K6) proceedingas the previous point we show that the function 119902 defined by(39) satisfies (H1) with 119892 instead of 119891 and 119883
119905instead of 119881
119905
and(H2) (i)-(ii)-(iii) with 120588 = 0Therefore assuming that 119892 isin 119861119904
119901119903(119872) with 119903 ge 1 119901 ge 2
and 119904 isin (0119873) or 119901 isin [1 2) and 119904 isin (1119901119873) Theorem 3proves the existance of a constant 119862 gt 0 satisfying
E (1003817100381710038171003817119892 minus 119892
10038171003817100381710038172
2) le 119862(
ln 119899
119899)
2119904(2119904+1)
(100)
Combining (94) (99) and (100) we end the proof ofTheorem 6
Conflict of Interests
The author declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
The author is thankful to the reviewers for their commentswhich have helped in improving the presentation
Journal of Probability and Statistics 11
References
[1] P DoukhanMixing Properties and Examples vol 85 of LectureNotes in Statistics Springer New York NY USA 1994
[2] M Carrasco and X Chen ldquoMixing and moment properties ofvarious GARCH and stochastic volatility modelsrdquo EconometricTheory vol 18 no 1 pp 17ndash39 2002
[3] R C Bradley Introduction to Strong Mixing Conditions Volume1 2 3 Kendrick Press 2007
[4] P M Robinson ldquoNonparametric estimators for time seriesrdquoJournal of Time Series Analysis vol 4 pp 185ndash207 1983
[5] G G Roussas ldquoNonparametric estimation inmixing sequencesof random variablesrdquo Journal of Statistical Planning and Infer-ence vol 18 no 2 pp 135ndash149 1987
[6] G G Roussas ldquoNonparametric regression estimation undermixing conditionsrdquo Stochastic Processes and their Applicationsvol 36 no 1 pp 107ndash116 1990
[7] Y K Truong and C J Stone ldquoNonparametric function estima-tion involving time seriesrdquo The Annals of Statistics vol 20 pp77ndash97 1992
[8] L H Tran ldquoNonparametric function estimation for time seriesby local average estimatorsrdquoThe Annals of Statistics vol 21 pp1040ndash1057 1993
[9] E Masry ldquoMultivariate local polynomial regression for timeseries uniform strong consistency and ratesrdquo Journal of TimeSeries Analysis vol 17 no 6 pp 571ndash599 1996
[10] E Masry ldquoMultivariate regression estimation local polynomialfitting for time seriesrdquo Stochastic Processes and their Applica-tions vol 65 no 1 pp 81ndash101 1996
[11] E Masry and J Fan ldquoLocal polynomial estimation of regressionfunctions for mixing processesrdquo Scandinavian Journal of Statis-tics vol 24 no 2 pp 165ndash179 1997
[12] D Bosq Nonparametric Statistics for Stochastic Processes Esti-mation and Prediction vol 110 of Lecture Notes in StatisticsSpringer New York NY USA 1998
[13] E Liebscher ldquoEstimation of the density and the regressionfunction under mixing conditionsrdquo Statistics and Decisions vol19 no 1 pp 9ndash26 2001
[14] D L Donoho and I M Johnstone ldquoIdeal spatial adaptation bywavelet shrinkagerdquo Biometrika vol 81 no 3 pp 425ndash455 1994
[15] D L Donoho and I M Johnstone ldquoAdapting to unknownsmoothness via wavelet shrinkagerdquo Journal of the AmericanStatistical Association vol 90 pp 1200ndash1224 1995
[16] D Donoho I Johnstone G Kerkyacharian and D PicardldquoWavelet shrinkage asymptopiardquo Journal of the Royal StatisticalSociety B vol 57 pp 301ndash369 1995
[17] D L Donoho I M Johnstone G Kerkyacharian and DPicard ldquoDensity estimation by wavelet thresholdingrdquo Annals ofStatistics vol 24 no 2 pp 508ndash539 1996
[18] A Antoniadis ldquoWavelets in statistics a review (with discus-sion)rdquo Journal of the Italian Statistical Society vol 6 pp 97ndash1441997
[19] W Hardle G Kerkyacharian D Picard and A TsybakovWavelet Approximation and Statistical Applications vol 129 ofLectures Notes in Statistics Springer New York NY USA 1998
[20] F Leblanc ldquoWavelet linear density estimator for a discrete-timestochastic process 119871
119901-lossesrdquo Statistics and Probability Letters
vol 27 no 1 pp 71ndash84 1996[21] K Tribouley andG Viennet ldquo119871
119901adaptive density estimation in
a alpha-mixing frameworkrdquo Annales de lrsquoinstitut Henri PoincareB vol 34 no 2 pp 179ndash208 1998
[22] E Masry ldquoWavelet-based estimation of multivariate regressionfunctions in besov spacesrdquo Journal of Nonparametric Statisticsvol 12 no 2 pp 283ndash308 2000
[23] P N Patil and Y K Truong ldquoAsymptotics for wavelet basedestimates of piecewise smooth regression for stationary timeseriesrdquo Annals of the Institute of Statistical Mathematics vol 53no 1 pp 159ndash178 2001
[24] H Doosti M Afshari and H A Niroumand ldquoWavelets fornonparametric stochastic regression with mixing stochasticprocessrdquo Communications in Statistics vol 37 no 3 pp 373ndash385 2008
[25] H Doosti and H A Niroumand ldquoMultivariate stochasticregression estimation by wavelet methods for stationary timeseriesrdquo Pakistan Journal of Statistics vol 27 no 1 pp 37ndash462009
[26] H Doosti M S Islam Y P Chaubey and P Gora ldquoTwo-dimensional wavelets for nonlinear autoregressive models withan application in dynamical systemrdquo Italian Journal of Pure andApplied Mathematics no 27 pp 39ndash62 2010
[27] J Cai and H Liang ldquoNonlinear wavelet density estimation fortruncated and dependent observationsrdquo International Journal ofWavelets Multiresolution and Information Processing vol 9 no4 pp 587ndash609 2011
[28] S Niu and H Liang ldquoNonlinear wavelet estimation of condi-tional density under left-truncated and 120572-mixing assumptionsrdquoInternational Journal of Wavelets Multiresolution and Informa-tion Processing vol 9 no 6 pp 989ndash1023 2011
[29] F Benatia and D Yahia ldquoNonlinear wavelet regression functionestimator for censored dependent datardquo Journal Afrika Statis-tika vol 7 no 1 pp 391ndash411 2012
[30] C Chesneau ldquoOn the adaptive wavelet deconvolution of adensity for strong mixing sequencesrdquo Journal of the KoreanStatistical Society vol 41 no 4 pp 423ndash436 2012
[31] C Chesneau ldquoWavelet estimation of a density in a GARCH-type modelrdquo Communications in Statistics vol 42 no 1 pp 98ndash117 2013
[32] C Chesneau ldquoOn the adaptive wavelet estimation of a mul-tidimensional regression function under alpha-mixing depen-dence beyond the standard assumptions on the noiserdquo Com-mentationes Mathematicae Universitatis Carolinae vol 4 pp527ndash556 2013
[33] Y P Chaubey and E Shirazi ldquoOn MISE of a nonlinear waveletestimator of the regression function based on biased data understrong mixing rdquo Communications in Statistics In press
[34] M Abbaszadeh andM Emadi ldquoWavelet density estimation andstatistical evidences role for a garch model in the weighteddistributionrdquo Applied Mathematics vol 4 no 2 pp 10ndash4162013
[35] B Delyon and A Juditsky ldquoOn minimax wavelet estimatorsrdquoApplied and Computational Harmonic Analysis vol 3 no 3 pp215ndash228 1996
[36] G Kerkyacharian and D Picard ldquoThresholding algorithmsmaxisets and well-concentrated basesrdquo Test vol 9 no 2 pp283ndash344 2000
[37] I Daubechies Ten Lectures on Wavelets SIAM 1992[38] A Cohen I Daubechies and P Vial ldquoWavelets on the interval
and fast wavelet transformsrdquo Applied and Computational Har-monic Analysis vol 1 no 1 pp 54ndash81 1993
[39] S Mallat A Wavelet Tour of Signal Processing The sparse waywith contributions fromGabriel Peyralphae ElsevierAcademicPress Amsterdam The Netherlands 3rd edition 2009
12 Journal of Probability and Statistics
[40] Y Meyer Ondelettes et Opalphaerateurs Hermann ParisFrance 1990
[41] V Genon-Catalot T Jeantheau and C Laredo ldquoStochasticvolatility models as hidden Markov models and statisticalapplicationsrdquo Bernoulli vol 6 no 6 pp 1051ndash1079 2000
[42] J Fan and J Koo ldquoWavelet deconvolutionrdquo IEEE Transactionson Information Theory vol 48 no 3 pp 734ndash747 2002
[43] A B Tsybakov Introduction alphaa lrsquoestimation non-paramalphaetrique Springer New York NY USA 2004
[44] E Masry ldquoStrong consistency and rates for deconvolutionof multivariate densities of stationary processesrdquo StochasticProcesses and their Applications vol 47 no 1 pp 53ndash74 1993
[45] R Kulik ldquoNonparametric deconvolution problem for depen-dent sequencesrdquo Electronic Journal of Statistics vol 2 pp 722ndash740 2008
[46] F Comte J Dedecker and M L Taupin ldquoAdaptive densitydeconvolution with dependent inputsrdquo Mathematical Methodsof Statistics vol 17 no 2 pp 87ndash112 2008
[47] H van Zanten and P Zareba ldquoA note on wavelet densitydeconvolution for weakly dependent datardquo Statistical Inferencefor Stochastic Processes vol 11 no 2 pp 207ndash219 2008
[48] W Hardle Applied Nonparametric Regression Cambridge Uni-versity Press Cambridge UK 1990
[49] V A Vasiliev ldquoOne investigation method of a ratios typeestimatorsrdquo in Proceedings of the 16th IFAC Symposium onSystem Identialphacation pp 1ndash6 Brussels Belgium July 2012(in progress)
[50] Y Davydov ldquoThe invariance principle for stationary processesrdquoTheory of Probability and Its Applications vol 15 no 3 pp 498ndash509 1970
[51] C Chesneau ldquoWavelet estimation via block thresholding aminimax study under 119871119901-riskrdquo Statistica Sinica vol 18 no 3pp 1007ndash1024 2008
[52] C Chesneau and H Doosti ldquoWavelet linear density estimationfor a GARCH model under various dependence structuresrdquoJournal of the Iranian Statistical Society vol 11 no 1 pp 1ndash212012
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
10 Journal of Probability and Statistics
where
119878 (119909)
=1
119892 (119909)(V (119909) minus V (119909)
+119891 (119909) (119892 (119909) minus 119892 (119909))) 1|119892(119909)|ge119888lowast2
119879 (119909) = 119891 (119909) 1|119892(119909)|lt119888lowast2
(90)
Using (K3) and the indicator function we have
|119878 (119909)| le 119862 (|V (119909) minus V (119909)| + 1003816100381610038161003816119892 (119909) minus 119892 (119909)1003816100381610038161003816) (91)
It follows from |119892(119909)| lt 119888lowast2 cap |119892(119909)| gt 119888
lowast sube |119892(119909) minus
119892(119909)| gt 119888lowast2 (K3) (K4) and the Markov inequality that
|119879 (119909)| le 1198621|119892(119909)minus119892(119909)|gt119888lowast2
le 1198621003816100381610038161003816119892 (119909) minus 119892 (119909)
1003816100381610038161003816 (92)
The triangular inequality yields
10038161003816100381610038161003816119891 (119909) minus 119891 (119909)10038161003816100381610038161003816 le 119862 (|V (119909) minus V (119909)| + 1003816100381610038161003816119892 (119909) minus 119892 (119909)
1003816100381610038161003816) (93)
The elementary inequality (119886 + 119887)2 le 2(1198862 + 1198872) implies that
E (10038171003817100381710038171003817119891 minus 119891
100381710038171003817100381710038172
2) le 119862 (E (V minus V2
2) + E (
1003817100381710038171003817119892 minus 11989210038171003817100381710038172
2)) (94)
We now bound this two MISEs via Theorem 3
Upper Bound for the MISE of V Under (K1)ndash(K6) thefunction 119902 defined by (38) satisfies the following
(H1) With V instead of 119891 since 1205851and 119883
1are independent
with E(1205851) = 0
E (119902 (120574119895119896
1198811))
= E (1198841120574119895119896
(1198831)) = E (119891 (119883
1) 120574
119895119896(119883
1))
= int1
0
119891 (119909) 120574119895119896
(119909) 119892 (119909) 119889119909 = int1
0
V (119909) 120574119895119896
(119909) 119889119909
(95)
(H2) (i)-(ii)-(iii) with 120588 = 0
(i) since 1198841(Ω) is bounded thanks to (K2) and (K3) say
|1198841| le 119872 with 119872 gt 0 we have
sup(119909119909lowast)isin1198811(Ω)
10038161003816100381610038161003816119902 (120574119895119896
(119909 119909lowast))
10038161003816100381610038161003816
= sup(119909119909lowast)isin[minus119872119872]times[01]
10038161003816100381610038161003816119909120574119895119896(119909
lowast)10038161003816100381610038161003816
le 119872 sup119909lowastisin[01]
10038161003816100381610038161003816120574119895119896(119909
lowast)10038161003816100381610038161003816 le 11986221198952
(96)
(ii) using the boundedness of 1198841(Ω) then (K4) we have
E (10038161003816100381610038161003816119902(120574119895119896
1198811)100381610038161003816100381610038162
) = E (1198842
1(120574
119895119896(119883
1))
2
)
le 119862E ((120574119895119896
(1198831))
2
)
= 119862int1
0
(120574119895119896
(119909))2
119892 (119909) 119889119909
le 119862int1
0
(120574119895119896
(119909))2
119889119909 le 119862
(97)
(iii) using the boundedness of 1198841(Ω) and making the
change of variables 119910 = 2119895119909 minus 119896 we obtain
int1198811(Ω)
10038161003816100381610038161003816119902 (120574119895119896
119909)10038161003816100381610038161003816 119889119909
= (int119872
minus119872
|119909| 119889119909)(int1
0
10038161003816100381610038161003816120574119895119896(119909
lowast)10038161003816100381610038161003816 119889119909lowast
)
= 1198722 int1
0
10038161003816100381610038161003816120574119895119896(119909)
10038161003816100381610038161003816 119889119909 le 1198622minus1198952
(98)
We conclude by applying Lemma 2 with 120588 = 0 (K5) and (K6)imply (F1) and the previous inequality implies (F2)
Therefore assuming that V isin 119861119904
119901119903(119872) with 119903 ge 1 119901 ge 2
and 119904 isin (0119873) or 119901 isin [1 2) and 119904 isin (1119901119873) Theorem 3proves the existence of a constant 119862 gt 0 satisfying
E (V minus V22) le 119862(
ln 119899
119899)
2119904(2119904+1)
(99)
Upper Bound for theMISE of 119892 Under (K1)ndash(K6) proceedingas the previous point we show that the function 119902 defined by(39) satisfies (H1) with 119892 instead of 119891 and 119883
119905instead of 119881
119905
and(H2) (i)-(ii)-(iii) with 120588 = 0Therefore assuming that 119892 isin 119861119904
119901119903(119872) with 119903 ge 1 119901 ge 2
and 119904 isin (0119873) or 119901 isin [1 2) and 119904 isin (1119901119873) Theorem 3proves the existance of a constant 119862 gt 0 satisfying
E (1003817100381710038171003817119892 minus 119892
10038171003817100381710038172
2) le 119862(
ln 119899
119899)
2119904(2119904+1)
(100)
Combining (94) (99) and (100) we end the proof ofTheorem 6
Conflict of Interests
The author declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
The author is thankful to the reviewers for their commentswhich have helped in improving the presentation
Journal of Probability and Statistics 11
References
[1] P DoukhanMixing Properties and Examples vol 85 of LectureNotes in Statistics Springer New York NY USA 1994
[2] M Carrasco and X Chen ldquoMixing and moment properties ofvarious GARCH and stochastic volatility modelsrdquo EconometricTheory vol 18 no 1 pp 17ndash39 2002
[3] R C Bradley Introduction to Strong Mixing Conditions Volume1 2 3 Kendrick Press 2007
[4] P M Robinson ldquoNonparametric estimators for time seriesrdquoJournal of Time Series Analysis vol 4 pp 185ndash207 1983
[5] G G Roussas ldquoNonparametric estimation inmixing sequencesof random variablesrdquo Journal of Statistical Planning and Infer-ence vol 18 no 2 pp 135ndash149 1987
[6] G G Roussas ldquoNonparametric regression estimation undermixing conditionsrdquo Stochastic Processes and their Applicationsvol 36 no 1 pp 107ndash116 1990
[7] Y K Truong and C J Stone ldquoNonparametric function estima-tion involving time seriesrdquo The Annals of Statistics vol 20 pp77ndash97 1992
[8] L H Tran ldquoNonparametric function estimation for time seriesby local average estimatorsrdquoThe Annals of Statistics vol 21 pp1040ndash1057 1993
[9] E Masry ldquoMultivariate local polynomial regression for timeseries uniform strong consistency and ratesrdquo Journal of TimeSeries Analysis vol 17 no 6 pp 571ndash599 1996
[10] E Masry ldquoMultivariate regression estimation local polynomialfitting for time seriesrdquo Stochastic Processes and their Applica-tions vol 65 no 1 pp 81ndash101 1996
[11] E Masry and J Fan ldquoLocal polynomial estimation of regressionfunctions for mixing processesrdquo Scandinavian Journal of Statis-tics vol 24 no 2 pp 165ndash179 1997
[12] D Bosq Nonparametric Statistics for Stochastic Processes Esti-mation and Prediction vol 110 of Lecture Notes in StatisticsSpringer New York NY USA 1998
[13] E Liebscher ldquoEstimation of the density and the regressionfunction under mixing conditionsrdquo Statistics and Decisions vol19 no 1 pp 9ndash26 2001
[14] D L Donoho and I M Johnstone ldquoIdeal spatial adaptation bywavelet shrinkagerdquo Biometrika vol 81 no 3 pp 425ndash455 1994
[15] D L Donoho and I M Johnstone ldquoAdapting to unknownsmoothness via wavelet shrinkagerdquo Journal of the AmericanStatistical Association vol 90 pp 1200ndash1224 1995
[16] D Donoho I Johnstone G Kerkyacharian and D PicardldquoWavelet shrinkage asymptopiardquo Journal of the Royal StatisticalSociety B vol 57 pp 301ndash369 1995
[17] D L Donoho I M Johnstone G Kerkyacharian and DPicard ldquoDensity estimation by wavelet thresholdingrdquo Annals ofStatistics vol 24 no 2 pp 508ndash539 1996
[18] A Antoniadis ldquoWavelets in statistics a review (with discus-sion)rdquo Journal of the Italian Statistical Society vol 6 pp 97ndash1441997
[19] W Hardle G Kerkyacharian D Picard and A TsybakovWavelet Approximation and Statistical Applications vol 129 ofLectures Notes in Statistics Springer New York NY USA 1998
[20] F Leblanc ldquoWavelet linear density estimator for a discrete-timestochastic process 119871
119901-lossesrdquo Statistics and Probability Letters
vol 27 no 1 pp 71ndash84 1996[21] K Tribouley andG Viennet ldquo119871
119901adaptive density estimation in
a alpha-mixing frameworkrdquo Annales de lrsquoinstitut Henri PoincareB vol 34 no 2 pp 179ndash208 1998
[22] E Masry ldquoWavelet-based estimation of multivariate regressionfunctions in besov spacesrdquo Journal of Nonparametric Statisticsvol 12 no 2 pp 283ndash308 2000
[23] P N Patil and Y K Truong ldquoAsymptotics for wavelet basedestimates of piecewise smooth regression for stationary timeseriesrdquo Annals of the Institute of Statistical Mathematics vol 53no 1 pp 159ndash178 2001
[24] H Doosti M Afshari and H A Niroumand ldquoWavelets fornonparametric stochastic regression with mixing stochasticprocessrdquo Communications in Statistics vol 37 no 3 pp 373ndash385 2008
[25] H Doosti and H A Niroumand ldquoMultivariate stochasticregression estimation by wavelet methods for stationary timeseriesrdquo Pakistan Journal of Statistics vol 27 no 1 pp 37ndash462009
[26] H Doosti M S Islam Y P Chaubey and P Gora ldquoTwo-dimensional wavelets for nonlinear autoregressive models withan application in dynamical systemrdquo Italian Journal of Pure andApplied Mathematics no 27 pp 39ndash62 2010
[27] J Cai and H Liang ldquoNonlinear wavelet density estimation fortruncated and dependent observationsrdquo International Journal ofWavelets Multiresolution and Information Processing vol 9 no4 pp 587ndash609 2011
[28] S Niu and H Liang ldquoNonlinear wavelet estimation of condi-tional density under left-truncated and 120572-mixing assumptionsrdquoInternational Journal of Wavelets Multiresolution and Informa-tion Processing vol 9 no 6 pp 989ndash1023 2011
[29] F Benatia and D Yahia ldquoNonlinear wavelet regression functionestimator for censored dependent datardquo Journal Afrika Statis-tika vol 7 no 1 pp 391ndash411 2012
[30] C Chesneau ldquoOn the adaptive wavelet deconvolution of adensity for strong mixing sequencesrdquo Journal of the KoreanStatistical Society vol 41 no 4 pp 423ndash436 2012
[31] C Chesneau ldquoWavelet estimation of a density in a GARCH-type modelrdquo Communications in Statistics vol 42 no 1 pp 98ndash117 2013
[32] C Chesneau ldquoOn the adaptive wavelet estimation of a mul-tidimensional regression function under alpha-mixing depen-dence beyond the standard assumptions on the noiserdquo Com-mentationes Mathematicae Universitatis Carolinae vol 4 pp527ndash556 2013
[33] Y P Chaubey and E Shirazi ldquoOn MISE of a nonlinear waveletestimator of the regression function based on biased data understrong mixing rdquo Communications in Statistics In press
[34] M Abbaszadeh andM Emadi ldquoWavelet density estimation andstatistical evidences role for a garch model in the weighteddistributionrdquo Applied Mathematics vol 4 no 2 pp 10ndash4162013
[35] B Delyon and A Juditsky ldquoOn minimax wavelet estimatorsrdquoApplied and Computational Harmonic Analysis vol 3 no 3 pp215ndash228 1996
[36] G Kerkyacharian and D Picard ldquoThresholding algorithmsmaxisets and well-concentrated basesrdquo Test vol 9 no 2 pp283ndash344 2000
[37] I Daubechies Ten Lectures on Wavelets SIAM 1992[38] A Cohen I Daubechies and P Vial ldquoWavelets on the interval
and fast wavelet transformsrdquo Applied and Computational Har-monic Analysis vol 1 no 1 pp 54ndash81 1993
[39] S Mallat A Wavelet Tour of Signal Processing The sparse waywith contributions fromGabriel Peyralphae ElsevierAcademicPress Amsterdam The Netherlands 3rd edition 2009
12 Journal of Probability and Statistics
[40] Y Meyer Ondelettes et Opalphaerateurs Hermann ParisFrance 1990
[41] V Genon-Catalot T Jeantheau and C Laredo ldquoStochasticvolatility models as hidden Markov models and statisticalapplicationsrdquo Bernoulli vol 6 no 6 pp 1051ndash1079 2000
[42] J Fan and J Koo ldquoWavelet deconvolutionrdquo IEEE Transactionson Information Theory vol 48 no 3 pp 734ndash747 2002
[43] A B Tsybakov Introduction alphaa lrsquoestimation non-paramalphaetrique Springer New York NY USA 2004
[44] E Masry ldquoStrong consistency and rates for deconvolutionof multivariate densities of stationary processesrdquo StochasticProcesses and their Applications vol 47 no 1 pp 53ndash74 1993
[45] R Kulik ldquoNonparametric deconvolution problem for depen-dent sequencesrdquo Electronic Journal of Statistics vol 2 pp 722ndash740 2008
[46] F Comte J Dedecker and M L Taupin ldquoAdaptive densitydeconvolution with dependent inputsrdquo Mathematical Methodsof Statistics vol 17 no 2 pp 87ndash112 2008
[47] H van Zanten and P Zareba ldquoA note on wavelet densitydeconvolution for weakly dependent datardquo Statistical Inferencefor Stochastic Processes vol 11 no 2 pp 207ndash219 2008
[48] W Hardle Applied Nonparametric Regression Cambridge Uni-versity Press Cambridge UK 1990
[49] V A Vasiliev ldquoOne investigation method of a ratios typeestimatorsrdquo in Proceedings of the 16th IFAC Symposium onSystem Identialphacation pp 1ndash6 Brussels Belgium July 2012(in progress)
[50] Y Davydov ldquoThe invariance principle for stationary processesrdquoTheory of Probability and Its Applications vol 15 no 3 pp 498ndash509 1970
[51] C Chesneau ldquoWavelet estimation via block thresholding aminimax study under 119871119901-riskrdquo Statistica Sinica vol 18 no 3pp 1007ndash1024 2008
[52] C Chesneau and H Doosti ldquoWavelet linear density estimationfor a GARCH model under various dependence structuresrdquoJournal of the Iranian Statistical Society vol 11 no 1 pp 1ndash212012
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Journal of Probability and Statistics 11
References
[1] P DoukhanMixing Properties and Examples vol 85 of LectureNotes in Statistics Springer New York NY USA 1994
[2] M Carrasco and X Chen ldquoMixing and moment properties ofvarious GARCH and stochastic volatility modelsrdquo EconometricTheory vol 18 no 1 pp 17ndash39 2002
[3] R C Bradley Introduction to Strong Mixing Conditions Volume1 2 3 Kendrick Press 2007
[4] P M Robinson ldquoNonparametric estimators for time seriesrdquoJournal of Time Series Analysis vol 4 pp 185ndash207 1983
[5] G G Roussas ldquoNonparametric estimation inmixing sequencesof random variablesrdquo Journal of Statistical Planning and Infer-ence vol 18 no 2 pp 135ndash149 1987
[6] G G Roussas ldquoNonparametric regression estimation undermixing conditionsrdquo Stochastic Processes and their Applicationsvol 36 no 1 pp 107ndash116 1990
[7] Y K Truong and C J Stone ldquoNonparametric function estima-tion involving time seriesrdquo The Annals of Statistics vol 20 pp77ndash97 1992
[8] L H Tran ldquoNonparametric function estimation for time seriesby local average estimatorsrdquoThe Annals of Statistics vol 21 pp1040ndash1057 1993
[9] E Masry ldquoMultivariate local polynomial regression for timeseries uniform strong consistency and ratesrdquo Journal of TimeSeries Analysis vol 17 no 6 pp 571ndash599 1996
[10] E Masry ldquoMultivariate regression estimation local polynomialfitting for time seriesrdquo Stochastic Processes and their Applica-tions vol 65 no 1 pp 81ndash101 1996
[11] E Masry and J Fan ldquoLocal polynomial estimation of regressionfunctions for mixing processesrdquo Scandinavian Journal of Statis-tics vol 24 no 2 pp 165ndash179 1997
[12] D Bosq Nonparametric Statistics for Stochastic Processes Esti-mation and Prediction vol 110 of Lecture Notes in StatisticsSpringer New York NY USA 1998
[13] E Liebscher ldquoEstimation of the density and the regressionfunction under mixing conditionsrdquo Statistics and Decisions vol19 no 1 pp 9ndash26 2001
[14] D L Donoho and I M Johnstone ldquoIdeal spatial adaptation bywavelet shrinkagerdquo Biometrika vol 81 no 3 pp 425ndash455 1994
[15] D L Donoho and I M Johnstone ldquoAdapting to unknownsmoothness via wavelet shrinkagerdquo Journal of the AmericanStatistical Association vol 90 pp 1200ndash1224 1995
[16] D Donoho I Johnstone G Kerkyacharian and D PicardldquoWavelet shrinkage asymptopiardquo Journal of the Royal StatisticalSociety B vol 57 pp 301ndash369 1995
[17] D L Donoho I M Johnstone G Kerkyacharian and DPicard ldquoDensity estimation by wavelet thresholdingrdquo Annals ofStatistics vol 24 no 2 pp 508ndash539 1996
[18] A Antoniadis ldquoWavelets in statistics a review (with discus-sion)rdquo Journal of the Italian Statistical Society vol 6 pp 97ndash1441997
[19] W Hardle G Kerkyacharian D Picard and A TsybakovWavelet Approximation and Statistical Applications vol 129 ofLectures Notes in Statistics Springer New York NY USA 1998
[20] F Leblanc ldquoWavelet linear density estimator for a discrete-timestochastic process 119871
119901-lossesrdquo Statistics and Probability Letters
vol 27 no 1 pp 71ndash84 1996[21] K Tribouley andG Viennet ldquo119871
119901adaptive density estimation in
a alpha-mixing frameworkrdquo Annales de lrsquoinstitut Henri PoincareB vol 34 no 2 pp 179ndash208 1998
[22] E Masry ldquoWavelet-based estimation of multivariate regressionfunctions in besov spacesrdquo Journal of Nonparametric Statisticsvol 12 no 2 pp 283ndash308 2000
[23] P N Patil and Y K Truong ldquoAsymptotics for wavelet basedestimates of piecewise smooth regression for stationary timeseriesrdquo Annals of the Institute of Statistical Mathematics vol 53no 1 pp 159ndash178 2001
[24] H Doosti M Afshari and H A Niroumand ldquoWavelets fornonparametric stochastic regression with mixing stochasticprocessrdquo Communications in Statistics vol 37 no 3 pp 373ndash385 2008
[25] H Doosti and H A Niroumand ldquoMultivariate stochasticregression estimation by wavelet methods for stationary timeseriesrdquo Pakistan Journal of Statistics vol 27 no 1 pp 37ndash462009
[26] H Doosti M S Islam Y P Chaubey and P Gora ldquoTwo-dimensional wavelets for nonlinear autoregressive models withan application in dynamical systemrdquo Italian Journal of Pure andApplied Mathematics no 27 pp 39ndash62 2010
[27] J Cai and H Liang ldquoNonlinear wavelet density estimation fortruncated and dependent observationsrdquo International Journal ofWavelets Multiresolution and Information Processing vol 9 no4 pp 587ndash609 2011
[28] S Niu and H Liang ldquoNonlinear wavelet estimation of condi-tional density under left-truncated and 120572-mixing assumptionsrdquoInternational Journal of Wavelets Multiresolution and Informa-tion Processing vol 9 no 6 pp 989ndash1023 2011
[29] F Benatia and D Yahia ldquoNonlinear wavelet regression functionestimator for censored dependent datardquo Journal Afrika Statis-tika vol 7 no 1 pp 391ndash411 2012
[30] C Chesneau ldquoOn the adaptive wavelet deconvolution of adensity for strong mixing sequencesrdquo Journal of the KoreanStatistical Society vol 41 no 4 pp 423ndash436 2012
[31] C Chesneau ldquoWavelet estimation of a density in a GARCH-type modelrdquo Communications in Statistics vol 42 no 1 pp 98ndash117 2013
[32] C Chesneau ldquoOn the adaptive wavelet estimation of a mul-tidimensional regression function under alpha-mixing depen-dence beyond the standard assumptions on the noiserdquo Com-mentationes Mathematicae Universitatis Carolinae vol 4 pp527ndash556 2013
[33] Y P Chaubey and E Shirazi ldquoOn MISE of a nonlinear waveletestimator of the regression function based on biased data understrong mixing rdquo Communications in Statistics In press
[34] M Abbaszadeh andM Emadi ldquoWavelet density estimation andstatistical evidences role for a garch model in the weighteddistributionrdquo Applied Mathematics vol 4 no 2 pp 10ndash4162013
[35] B Delyon and A Juditsky ldquoOn minimax wavelet estimatorsrdquoApplied and Computational Harmonic Analysis vol 3 no 3 pp215ndash228 1996
[36] G Kerkyacharian and D Picard ldquoThresholding algorithmsmaxisets and well-concentrated basesrdquo Test vol 9 no 2 pp283ndash344 2000
[37] I Daubechies Ten Lectures on Wavelets SIAM 1992[38] A Cohen I Daubechies and P Vial ldquoWavelets on the interval
and fast wavelet transformsrdquo Applied and Computational Har-monic Analysis vol 1 no 1 pp 54ndash81 1993
[39] S Mallat A Wavelet Tour of Signal Processing The sparse waywith contributions fromGabriel Peyralphae ElsevierAcademicPress Amsterdam The Netherlands 3rd edition 2009
12 Journal of Probability and Statistics
[40] Y Meyer Ondelettes et Opalphaerateurs Hermann ParisFrance 1990
[41] V Genon-Catalot T Jeantheau and C Laredo ldquoStochasticvolatility models as hidden Markov models and statisticalapplicationsrdquo Bernoulli vol 6 no 6 pp 1051ndash1079 2000
[42] J Fan and J Koo ldquoWavelet deconvolutionrdquo IEEE Transactionson Information Theory vol 48 no 3 pp 734ndash747 2002
[43] A B Tsybakov Introduction alphaa lrsquoestimation non-paramalphaetrique Springer New York NY USA 2004
[44] E Masry ldquoStrong consistency and rates for deconvolutionof multivariate densities of stationary processesrdquo StochasticProcesses and their Applications vol 47 no 1 pp 53ndash74 1993
[45] R Kulik ldquoNonparametric deconvolution problem for depen-dent sequencesrdquo Electronic Journal of Statistics vol 2 pp 722ndash740 2008
[46] F Comte J Dedecker and M L Taupin ldquoAdaptive densitydeconvolution with dependent inputsrdquo Mathematical Methodsof Statistics vol 17 no 2 pp 87ndash112 2008
[47] H van Zanten and P Zareba ldquoA note on wavelet densitydeconvolution for weakly dependent datardquo Statistical Inferencefor Stochastic Processes vol 11 no 2 pp 207ndash219 2008
[48] W Hardle Applied Nonparametric Regression Cambridge Uni-versity Press Cambridge UK 1990
[49] V A Vasiliev ldquoOne investigation method of a ratios typeestimatorsrdquo in Proceedings of the 16th IFAC Symposium onSystem Identialphacation pp 1ndash6 Brussels Belgium July 2012(in progress)
[50] Y Davydov ldquoThe invariance principle for stationary processesrdquoTheory of Probability and Its Applications vol 15 no 3 pp 498ndash509 1970
[51] C Chesneau ldquoWavelet estimation via block thresholding aminimax study under 119871119901-riskrdquo Statistica Sinica vol 18 no 3pp 1007ndash1024 2008
[52] C Chesneau and H Doosti ldquoWavelet linear density estimationfor a GARCH model under various dependence structuresrdquoJournal of the Iranian Statistical Society vol 11 no 1 pp 1ndash212012
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
12 Journal of Probability and Statistics
[40] Y Meyer Ondelettes et Opalphaerateurs Hermann ParisFrance 1990
[41] V Genon-Catalot T Jeantheau and C Laredo ldquoStochasticvolatility models as hidden Markov models and statisticalapplicationsrdquo Bernoulli vol 6 no 6 pp 1051ndash1079 2000
[42] J Fan and J Koo ldquoWavelet deconvolutionrdquo IEEE Transactionson Information Theory vol 48 no 3 pp 734ndash747 2002
[43] A B Tsybakov Introduction alphaa lrsquoestimation non-paramalphaetrique Springer New York NY USA 2004
[44] E Masry ldquoStrong consistency and rates for deconvolutionof multivariate densities of stationary processesrdquo StochasticProcesses and their Applications vol 47 no 1 pp 53ndash74 1993
[45] R Kulik ldquoNonparametric deconvolution problem for depen-dent sequencesrdquo Electronic Journal of Statistics vol 2 pp 722ndash740 2008
[46] F Comte J Dedecker and M L Taupin ldquoAdaptive densitydeconvolution with dependent inputsrdquo Mathematical Methodsof Statistics vol 17 no 2 pp 87ndash112 2008
[47] H van Zanten and P Zareba ldquoA note on wavelet densitydeconvolution for weakly dependent datardquo Statistical Inferencefor Stochastic Processes vol 11 no 2 pp 207ndash219 2008
[48] W Hardle Applied Nonparametric Regression Cambridge Uni-versity Press Cambridge UK 1990
[49] V A Vasiliev ldquoOne investigation method of a ratios typeestimatorsrdquo in Proceedings of the 16th IFAC Symposium onSystem Identialphacation pp 1ndash6 Brussels Belgium July 2012(in progress)
[50] Y Davydov ldquoThe invariance principle for stationary processesrdquoTheory of Probability and Its Applications vol 15 no 3 pp 498ndash509 1970
[51] C Chesneau ldquoWavelet estimation via block thresholding aminimax study under 119871119901-riskrdquo Statistica Sinica vol 18 no 3pp 1007ndash1024 2008
[52] C Chesneau and H Doosti ldquoWavelet linear density estimationfor a GARCH model under various dependence structuresrdquoJournal of the Iranian Statistical Society vol 11 no 1 pp 1ndash212012
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
top related