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Empirical Likelihood Inference for NonparametricRegression Functions with Functional StationaryErgodic DataXian-zhu Xiong a b & Zheng-yan Lin aa Department of Mathematics , Zhejiang University , Hangzhou , Chinab College of Mathematics and Computer Science , Fuzhou University , Fuzhou , ChinaAccepted author version posted online: 22 May 2013.Published online: 20 Aug 2013.

To cite this article: Xian-zhu Xiong & Zheng-yan Lin (2013) Empirical Likelihood Inference for Nonparametric RegressionFunctions with Functional Stationary Ergodic Data, Communications in Statistics - Theory and Methods, 42:19, 3421-3431, DOI:10.1080/03610926.2011.630766

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Communications in Statistics—Theory and Methods, 42: 3421–3431, 2013Copyright © Taylor & Francis Group, LLCISSN: 0361-0926 print/1532-415X onlineDOI: 10.1080/03610926.2011.630766

Empirical Likelihood Inference for NonparametricRegression Functions with Functional Stationary

Ergodic Data

XIAN-ZHU XIONG1�2 AND ZHENG-YAN LIN1

1Department of Mathematics, Zhejiang University, Hangzhou, China2College of Mathematics and Computer Science, Fuzhou University,Fuzhou, China

In this article, we employ the empirical likelihood method to construct confidenceintervals for nonparametric regression functions with functional stationary ergodicdata. Under some wild conditions, it is proved that the empirical likelihood ratiostatistic is asymptotically chi-squared distributed.

Keywords Confidence interval; Empirical likelihood; N-W estimator; Stationaryergodic data.

Mathematics Subject Classification 62G08; 62G15.

1. Introduction

The nonparametric regression techniques in modeling economic and financial timeseries data have gained a lot of attention during the last two decades. Bothestimation and specification testing have been systematically examined for real-valued i.i.d. or weakly dependent stationary processes; see, for example, Gyorfi et al.(1989), Robinson (1989), Yoshihara (1994), Härdle et al. (1997), Fan and Yao (2003)and the references therein.

Recently, there has been an increasing interest in the study of functional data.For an overview of the present state on nonparametric functional data analysis,we refer to the works of Ferraty and Vieu (2000, 2004), Ferraty et al. (2002),Masry (2005), Ferraty et al. (2007), Delsol (2009), Laib and Louani (2010), andthe recent monograph by Ferraty and Vieu (2006), and the references therein.For more general recent overviews on functional data analysis, including not onlythese nonparametric points of view, the reader may have a look at Ramsay andSilverman (1997, 2002), the recent handbook by Ferraty and Romain (2011), aswell as the various special issues of high level statistical journals that have been

Received January 17, 2011; Accepted October 5, 2011Address correspondence to Zheng-Yan Lin, Department of Mathematics, Zhejiang

University, Hangzhou 310027, People’s Republic of China; E-mail: zlin@zju.edu.cn

3421

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3422 Xiong and Lin

recently devoted to this problem (see Davidian et al., 2004; Ferraty, 2010; González-Manteiga and Vieu, 2007; Valderrama, 2007).

To formulate the functional regression estimation problem, let �Xi� Yi�i∈� be asequence of pairs of random elements where Yi is a real-valued random variable andXi takes its values in some semi-metric abstract space ��� d�·� ·��. This covers thecase of semi-normed spaces of possibly infinite dimension (e.g., Hilbert or Banachspaces) with the norm � · � and the distance d�x� y� = �x − y�. Assume, for k = 1� 2,that E��Y1�k� < � and that, for a fixed x ∈ �, the conditional mean function r�x� �=E�Y1 �X1 = x� and the conditional variance W2�x� �= E��Y1 − r�X1��

2 �X1 = x� exist.The N-W estimator of r�x� has been introduced by Ferraty and Vieu (2000). It isdefined by

r̂n�x� =∑n

i=1 YiK(

d�x�Xi�

h

)∑n

i=1 K(

d�x�Xi�

h

) �= r̂n�2�x�

r̂n�1�x�(1.1)

when the denominator is not equal to 0. Here, K is a real-valued kernel function,h �= hn is the bandwith parameter (which goes to 0 as n goes to infinity) and

r̂n�j�x� =1

nE��1�x��

n∑i=1

Yj−1i �i�x� for j = 1� 2� (1.2)

where

�i�x� = K

(d�x�Xi�

h

)�

In the independent functional data case, the asymptotic results of the estimatorr̂n�x� including the mean squared convergence, with rates, as well as the asymptoticnormality, were obtained by Ferraty et al. (2007). When mixing processes areconsidered, rates of almost sure uniform convergence, over a compact set of r̂n�x�were established in Ferraty and Vieu (2004) while Masry (2005) obtained the meansquared convergence and asymptotic normality. Ferraty et al. (2002) and Delsol(2009) considered the process prediction problem in the strongly mixing dependencestructure.

However, the mixing properties of a number of well-known processes arestill open questions (refer to Andrews, 1984; Chernick, 1981). In particular, theprocess Xi = Xi−1 + i, where ∈ �0� 1/2� and �i�i∈� is a sequence of independentBernoulli random variables, is not strongly mixing since the mixing coefficient�n = 1/4 for every n ∈ �. To be more convenient towards a number of practicalapplications, in the ergodic data setting, Laib and Louani (2010) considered theregression function estimation when the data are functional stationary ergodic. Theyestablished the consistency in probability of the estimator r̂n�x�, with a rate, as wellas the asymptotic normality by the martingale approximation method. They alsoconstructed confidence intervals of r�x� by the normal approximation method (seeCorollary 1 in Laib and Louani, 2010). In addition, they pointed out that the poorconvergence properties can be avoided by using the functional version of the usualwild bootstrap proposed by Ferraty et al. (2007). However, an obvious shortcomingof the method is that confidence intervals are always symmetric, artificially centered

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Empirical Likelihood for Functional Ergodic Data 3423

at the point functional estimate, thus unable to provide more information ondirections of variability of the underlying estimators. Moreover, the asymptoticvariances are usually estimated with large biases.

In this article, we employ the empirical likelihood method to constructconfidence intervals for r�x�. One of the motivations is that the empiricallikelihood method does not involve any variance estimation, the other is that theempirical likelihood-based confidence interval is shaped “automatically” by thesample without the the predetermined symmetry like the normal approximationmethod. The empirical likelihood method was introduced by Owen (1988) and itsgeneral property was studied by Owen (1990). Hall (1990) discussed the pseudo-likelihood theory for the method. DiCiccio et al. (1991) proved that the empiricallikelihood is Bartlett-correctable and thus it has an advantage over another popularnonparametric method for establishing confidence intervals, the bootstrap. Qin andLawless (1994) gave a general account on the equivalence between the empiricallikelihood and the method of estimating equations. In addition to the general theoryof the method, its various applications have also been studied by many authors,e.g., linear models (Chen, 1994; Owen, 1991), quantiles (Chen and Hall, 1993),generalized linear models (Kolaczyk, 1994), partial linear models (Wang and Jing,2003), diffusion processes (Lin and Wang, 2010; Xu, 2009), among others.

The purpose of this article is to establish the Wilks phenomenon for empiricallikelihood in the nonparametric function regression with functional stationaryergodic data. The rest of this article is organized as follows. In Sec. 2, weintroduce the empirical likelihood inference for the nonparametric regressionfunction, required notations, and assumptions. The main result, Theorem 2.1, is alsostated in the same section. In Sec. 3, Theorem 2.1 will be proved. The conclusionspresented in Sec. 4 contain some ideas for further research.

2. The Methodology and the Result

Firstly, we describe the empirical likelihood method in constructing confidenceintervals for r�x�. Let gri�x� h� �� = �i�x��Yi − ��. Thus, the N-W estimator r̂n�x�satisfies the estimating equation

n∑i=1

gri�x� h� r̂n�x�� = 0� (2.1)

For a candidate value � of the target quantity r�x�, the EL ratio statistic is defined as

Rr�x� h� �� = max�p1�p2�����pn�

{n∏

i=1

npi

∣∣∣∣n∑

i=1

pigri�x� h� �� = 0� pi ≥ 0�n∑

i=1

pi = 1

}� (2.2)

By the method of Lagrange multipliers, one can obtain pi = 1/�n�1+ gri�x� h� ����,where satisfies

n∑i=1

gri�x� h� ��

1+ gri�x� h� ��= 0� (2.3)

Let lr�x� h� �� = −2 logRr�x� h� �� = 2∑n

i=1 log�1+ gri�x� h� ���.

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3424 Xiong and Lin

Now, we introduce some notations. Let �i = ���X1� Y1�� � � � � �Xi� Yi��, �i =���X1� Y1�� � � � � �Xi� Yi�� Xi+1� and B�x� u� = �y ∈ � �d�x� y� ≤ u�. Let Fx�u� = P�d�x�Xi� ≤ u� = P�Xi ∈ B�x� u�� and F�i−1

x �u� = P�d�x�Xi� ≤ u ��i−1� = P�Xi ∈ B�x� u� ��i−1�. From now on, set

r̄n�j�x� =n∑

i=1

E�Yj−1i �i�x� ��i−1� for j = 1� 2� (2.4)

To obtain the asymptotic distribution of lr�x� h� r�x��, we need the followingassumptions.

(A1) K is a non negative bounded kernel of class �1 over its support [0, 1]and K�1� > 0. The derivative K′ exists on [0, 1] and satisfies the condition K′�t� <0�∀ t ∈ �0� 1� and � ∫ 1

0 K′�u�du� < �.

(A2) There exist a sequence of nonnegative bounded random functions�fi�1�x��i≥1, a sequence of random functions �gi�x�u��i≥1, a deterministic nonnegativebounded function f1�x� and a non negative real function ��u� tending to 0, as itsvariate tends to 0, such that:

(i) Fx�u� = ��u�f1�x�+ o���u�� as u → 0;(ii) for any i ∈ �� F�i−1

x �u� = ��u�fi�1�x�+ gi�x�u� with gi�x�u� = oa�s����u�� asu → 0, gi�x�u�/��u� almost surely bounded and n−1∑n

i=1 gji�x�u� = oa�s���

j�u�� asn → �, for j = 1� 2;

(iii) n−1∑ni=1 f

ji�1�x� → f

j1 �x�� almost surely as n → �� for j = 1� 2; and

(iv) there exists a non decreasing bounded function �0 such that, uniformly in u ∈�0� 1�,

��hu�

��h�= �0�u�+ o�1�� as h → 0 and

∣∣∣∣∫ 1

0K′�u��0�u�du

∣∣∣∣ < ��

(A3) (i) The conditional mean of Yi given �i−1 depends only on Xi� i.e., forany i ≥ 1� E�Yi ��i−1� = r�Xi� almost surely.

(ii) The conditional variance of Yi given �i−1 depends only on Xi, i.e., for any i ≥1� E��Yi − r�Xi��

2 ��i−1� = W2�Xi� almost surely. Moreover, the function W2 iscontinuous in a neighborhood of x.

(A4) (i) �r�u�− r�v�� ≤ c1d�u� v�� for all �u� v� ∈ �2 and some � > 0 and a

constant c1 > 0.(ii) E�Y 4

1 � < � and the function W 4�u� �= E��Yi − r�x��4�Xi = u�� u ∈ � is continuousin a neighborhood of x.

(A5) n��h� → �� as n → �.

Remark 2.1.

(a) Conditions used here are just the same as conditions (A1)–(A4) in Laib andLouani (2010). In fact, condition (A1) implies � ∫ 1

0 �Kj�′�u�du� < � for j ≥ 1,

conditions (A1) and (A2)(iv) imply � ∫ 10 �K

j�′�u��0�u�du� < � for j ≥ 1, andcondition (A4)(ii) is similar to condition (A5)(ii) in Laib and Louani (2010)which is used to establish Corollary 1 to construct confidence intervals for r�x�.Hypothesis (A5) is standard in nonparametric regression with functional data.

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Empirical Likelihood for Functional Ergodic Data 3425

(b) As discussed in Laib and Louani (2010), conditions (A1) and (A4)(i) are verycommon and usual in nonparametric functional estimation. Conditions (A2)contain the ergodic nature of the data and the small ball techniques used in thisarticle. They play an important role in the ergodic and functional context of thisarticle. Various situations where condition (A2)(ii) is satisfied are considered inLaib and Louani (2010).

The mail result of this article is then as follows.

Theorem 2.1. Suppose that conditions (A1)–(A5) hold and that h��n��h��1/2 → 0 asn → �. Then,

lr�x� h� r�x��d−→ �2�1�� (2.5)

Remark 2.2. Let z� satisfy P��2�1� ≤ z�� = 1− � with 0 < � < 1. It follows fromTheorem 2.1 that an empirical likelihood-based confidence interval for r�x� withasymptotically correct coverage probability 1− � can be constructed as

I� = �� � lr�x� h� �� ≤ z��� (2.6)

3. Proof of Theorem 2.1

Firstly, we introduce some useful lemmas and corollaries.

Lemma 3.1 (Laib and Louani, 2010). Assume that conditions (A1), (A2)(i), (A2)(ii),and (A2)(iv) hold. For any real numbers j ≥ 1 and k ≥ 1, as n → �, we have:

(i) 1��h�

E��ji �x� ��i−1� = Mjfi�1�x�+ Oa�s��

gi�x�h�

��h��;

(ii) 1��h�

E��j1�x�� = Mjf1�x�+ o�1�;

(iii) 1�k�h�

�E��1�x���k = Mk

1fk1 �x�+ o�1�,

where Mj = Kj�1�− ∫ 10 �K

j�′�u��0�u�du.

Lemma 3.2 (Laib and Louani, 2010). Assume that conditions (A1), (A2), and (A5)

hold. Then r̂n�1�x�P−→ 1.

Lemma 3.3 (Laib and Louani, 2010). Under the conditions of Theorem 2.1,

√n��h��r̂n�x�− r�x��

d−→ N�0� �2�x���

where �2�x� = M2W2�x�

M21 f1�x�

.

Corollary 3.1. Under the conditions of Lemma 3.3,∑n

i=1 gri�x� h� r�x�� = Op�√n��h��.

Proof. Lemma 3.3 implies√n��h�

∑ni=1 gri�x�h�r�x��∑n

i=1 �i�x�

d−→ N�0� �2�x��. Combined with

Lemma 3.2, Lemma 3.3 implies∑n

i=1 gri�x�h�r�x��√n��h�· E��1�x��

��h�

d−→ N�0� �2�x��. Then∑n

i=1 gri�x�h�r�x��√n��h�· E��1�x��

��h�

=Op�1�. Again, according to Lemma 3.1(ii), we have

∑ni=1 gri�x� h� r�x�� = Op

�√n��h��.

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3426 Xiong and Lin

Lemma 3.4. Let Ar�x� h� �� = �∑n

i=1 �i�x��−2∑n

i=1 g2ri�x� h� ��. Assume that conditions

(A1), (A2), (A4)(ii), and (A5) hold. Then n��h�Ar�x� h� r�x��P−→ �2�x�.

Proof.

n��h�Ar�x� h� r�x�� =[nE��1�x��∑n

i=1 �i�x�

]2· ��h�

nE2��1�x��

n∑i=1

�Yi − r�x��2�2i �x�� (3.1)

Let Zi = �Yi − r�x��2�2i �x�� 1 ≤ i ≤ n, then

��h�

nE2��1�x��

n∑i=1

�Yi − r�x��2�2i �x� =

��h�

nE2��1�x��

n∑i=1

Zi = T1�n�x�+ T2�n�x�� (3.2)

where T1�n�x� = ��h�

nE2��1�x��

∑ni=1�Zi − E�Zi ��i−1��� T2�n�x� = ��h�

nE2��1�x��

∑ni=1 E�Zi ��i−1�.

From (3.12) in Laib and Louani (2010), we have

T2�n�x�P−→ �2�x�� (3.3)

For any ,

P��T1�n�x�� > � ≤ E �∑n

i=1�Zi − E�Zi ��i−1���2

2· �2�h�

n2E4��1�x��

≤ c2E[∑n

i=1�Zi − E�Zi ��i−1��2]

2· �2�h�

n2E4��1�x��

(by Burkholder’s inequality)

≤ c3�2�h�E�Z2

1�

n2E4��1�x���by Jensen’s inequality �

≤ c3�2�h�E��4

1�x�E��Y1 − r�x��4 �X1��

n2E4��1�x��

≤ c3�2�h��W 4�x�+ o�1��E��4

1�x��

n2E4��1�x��(by assumption(A4)(ii))

≤ c3�W 4�x�+ o�1��E��41�x��

��h�

n2 E4��1�x��

�4�h�· ��h�

≤ c3�W 4�x�+ o�1���M4f1�x�+ o�1��

n��h�2�M41f

41 �x�+ o�1��

�by Lemma 3.1�

−→ 0� as n −→ ��

Then

T1�n�x�P−→ 0� (3.4)

Lemma 3.4 follows from (3.1)–(3.4) and Lemma 3.2.

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Empirical Likelihood for Functional Ergodic Data 3427

Corollary 3.2. Under the conditions of Lemma 3.4,

n∑i=1

g2ri�x� h� r�x�� = M2W2�x�f1�x�n��h�+ n��h�op�1��

Proof. From (3.2) and the proof of Lemma 3.4, we have

��h�

nE2��1�x��

n∑i=1

g2ri�x� h� r�x��P−→ �2�x��

that is∑n

i=1 g2ri�x�h�r�x��

n��h�E2 ��1�x��

�2�h�

P−→ �2�x�. According to Lemma 3.1(iii),∑n

i=1 g2ri�x�h�r�x��

n��h�

P−→M2W2�x�f1�x�. Then Corollary 3.2 follows.

Lemma 3.5. Assume that conditions (A1), (A2), (A4)(ii), and (A5) hold. Then,

max1≤i≤n

�gri�x� h� r�x��� = op�√n��h���

Proof. For any ,

P�max1≤i≤n

�gri�x� h� r�x��� > √n��h��

≤ nP���Y1 − r�x���1�x��4 > 4�n��h��2�

≤ nE��41�x�E��Y1 − r�x��4�X1��

4�n��h��2

= �W 4�x�+ o�1��E��41�x��

n4�2�h�

= �W 4�x�+ o�1���M4f1�x�+ o�1��n��h�4

�by Lemma 3.1(ii)�

−→ 0� as n −→ ��

Then Lemma 3.5 follows.

Proof of Theorem 2.1. (1) Firstly, it will be shown that

= Op

(1√

n��h�

)� (3.5)

Let gjr�x� h� r�x�� =∑n

i=1 gjri�x� h� r�x��� j = 1� 2.

From (2.3),

0 =∣∣∣∣∣

n∑i=1

gri�x� h� r�x��

1+ gri�x� h� r�x��

∣∣∣∣∣=∣∣∣∣∣

n∑i=1

g2ri�x� h� r�x��

1+ gri�x� h� r�x��−

n∑i=1

gri�x� h� r�x��

∣∣∣∣∣

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3428 Xiong and Lin

≥n∑

i=1

� �g2ri�x� h� r�x��1+ gri�x� h� r�x��

−∣∣∣∣∣

n∑i=1

gri�x� h� r�x��

∣∣∣∣∣(1+ gri�x� h� r�x�� =

1npi

> 0)

≥ � �g2r �x� h� r�x��1+ � �max1≤i≤n �gri�x� h� r�x���

− �g1r �x� h� r�x����

Then � ��g2r �x� h� r�x��−max1≤i≤n �gri�x� h� r�x���g1r �x� h� r�x���� ≤ �g1r �x� h� r�x���.By Corollary 3.1, Corollary 3.2, and Lemma 3.5,

� ��M2W2�x�f1�x�n��h�+ n��h�op�1�− op�√n��h��Op�

√n��h��� ≤ Op�

√n��h���

Then = Op

(1√n��h�

).

(2) Now it will be shown that

=(

n∑i=1

g2ri�x� h� r�x��

)−1 (n∑

i=1

gri�x� h� r�x��

)+ op

(1√

n��h�

)� (3.6)

From (2.3),

0 =n∑

i=1

gri�x� h� r�x��

1+ gri�x� h� r�x��

=n∑

i=1

gri�x� h� r�x���1− gri�x� h� r�x��+ 2g2ri�x� h� r�x��

1+ gri�x� h� r�x��

=n∑

i=1

gri�x� h� r�x��− n∑

i=1

g2ri�x� h� r�x��+n∑

i=1

2g3ri�x� h� r�x��

1+ gri�x� h� r�x��� (3.7)

Let �i = gri�x� h� r�x��. Then by Lemma 3.5 and (3.5),

max1≤i≤n

��i� = Op

(1√

n��h�

)op�√n��h�� = op�1�� (3.8)

The final term in (3.7) is bounded by

2 max1≤i≤n

�gri�x� h� r�x���n∑

i=1

g2ri�x� h� r�x��

�1+ �i�

= Op

(1

n��h�

)op

(√n��h�

)�M2W2�x�f1�x�n��h�+ n��h�op�1��Op�1�

= op�√n��h���

by Corollary 3.2, Lemma 3.5, (3.5), and (3.8). Then (3.6) follows from (3.7) andCorollary 3.2.

(3) By Taylor’s expansion of the logarithmic function,

log�1+ gri�x� h� r�x��� = gri�x� h� r�x��−12 2g2ri�x� h� r�x��+ �ri�r�x���

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Empirical Likelihood for Functional Ergodic Data 3429

where �ri�r�x�� satisfies that for some finite number M > 0,

P���ri�r�x��� ≤ M� gri�x� h� r�x���3� → 1� ∀ 1 ≤ i ≤ n�

By Corollary 3.2, Lemma 3.5, and (3.5),

n∑i=1

� gri�x� h� r�x���3 ≤ � �3 sup1≤i≤n

�gri�x� h� r�x���n∑

i=1

g2ri�x� h� r�x��

= Op

(1

�n��h��3/2

)op�√n��h��Op�n��h�� = op�1��

Then,

lr�x� h� r�x�� = 2n∑

i=1

log�1+ gri�x� h� r�x���

= 2 n∑

i=1

gri�x� h� r�x��− 2n∑

i=1

g2ri�x� h� r�x��+ op�1�

=(

n∑i=1

g2ri�x� h� r�x��

)−1 (n∑

i=1

gri�x� h� r�x��

)2

+ op�1�

(by (3.6) and Corollary 3.2)

= A−1r �x� h� r�x��B2

r �x� h� r�x��+ op�1�� (3.9)

where Br�x� h� r�x�� =(∑n

i=1 �i�x�)−1∑n

i=1 gri�x� h� r�x��. Lemma 3.3 implies

√n��h�Br�x� h� r�x��

d−→ N�0� �2�x��� (3.10)

Theorem 2.1 follows from (3.9), (3.10), and Lemma 3.4.

4. Conclusions and Perspectives

This article provides a theoretical framework about empirical likelihood inferencefor the nonparametric regression function with functional stationary ergodic data.To prove the results, the methodology is based upon the martingale approximationused in Laib and Louani (2010). And hypotheses (A2) play an important role, whichinvolve the ergodic nature of the data and the small ball techniques.

Theorem 2.1 can be used to obtain empirical likelihood-based confidenceintervals for the nonparametric regression function without a secondary varianceestimation. The main issue to apply Theorem 2.1 consists in bandwidthselection under the condition that as n → �� h��n��h��1/2 → 0, that is, as n → �,h��nFx�h��

1/2 → 0. The bandwidth h is relevant to the expression of the theoreticaldistribution Fx�h�, however, in the finite dimensional context, the bandwidth h isnot �h = o�n−1/�4+d��, where d is the dimension of X; see Chen and Van Keilegom(2009). Hence, the solution of the problem depends on the ability of approximatingthe distribution. For a fixed sample, as pointed out by a referee, bootstraping may

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3430 Xiong and Lin

be useful for the problem which was used in Ferraty et al. (2010) and González-Manteiga and Martinez-Calvo (2011). Therefore, the natural prospects of this articleare to implement the bootstrap method and study its practical and theoreticalbehavior on the nonparametric functional distribution.

Recently, Benhenni et al. (2008) studied the nonparametric regression estimatorwith functional data under long memory conditions. They established the pointwiseand uniform convergence of the N-W estimator. Their results may be an extensionof the results by Ferraty and Vieu (2004) for short memory (under strongly mixingconditions). Long memory and ergodic theory are two nice way for relaxingthe strongly mixing assumptions. It is obvious the a strongly mixing stationaryprocess is ergodic, and there are some examples of ergodic but not strongly mixingstationary processes. In addition, the commonly used condition on the stronglymixing coefficient(see condition (8) in Ferraty and Vieu, 2004) can make sure thatthe strongly mixing stationary process is short memory, that is, not long memory.Finally, many stationary and mixing processes are also ergodic. For example, ifyou want to prove the beta-mixing dependence, you only need to prove that it isgeometrically ergodic. For the long memory case, under some conditions, it alsomay be ergodic. Hence, another prospect of this article is to consider the empiricallikelihood inference for the nonparametric regression function with functional dataunder long memory conditions.

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