estimation with univa.ria.te mixed case interval censored data · icm algorithm, and jongbloed...
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Estimation with Univa.ria.te "Mixed Case" IntervalCensored Data
by
Shuguang SongThe Boeing Company
TECHNICAL REPORT No. 414
August, 2002
Department of Statistics
Box 354322
University of \Vashington
Seattle, Washington, 98195 USA
Estimation with Univariate "Mixed Case" IntervalCensored Data
Shuguang Song
The Boeing Company
Abstract: In this paper, we study the Nonparametric Maximum Likelihood Estimator (NPMLE)of univariate "Mixed Case" interval-censored data in which the number of observation times, andobservation time themselves are random variables. We provide a characterization of the NPMLE,then formulate the ICM algorithm for computing the NPMLE. We also study the asymptoticproperties of the NPMLE: consistency, global rates of convergence with and without a separationcondition, an asymptotic minimax lower bound and pointwise asymptotic distribution for the "toyestimator" proposed by GROENEBOOM AND WELLNER (1992)
Key words and phrases: convex minorant algorithm, maximum likelihood, empirical processes,interval censored data, consistency, rate of convergence, asymptotic distribution.
1 Introduction and Formulation of the Problem
Univariate interval censoring problem arises when a random variable, such as failure time or onsetof disease, can not be directly observed, but is only known to be in an interval determined by severalobservation times. Interval censored data and the corresponding application of statistical methodscan be found in animal carcinogenicity, epidemiology, HIV and AIDS studies, see FINKELSTEIN ANDWOLFE (1985), FINKELSTEIN (1986), SELF AND GROSSMAN (1986), BECKER AND MELBYE (1991) andARAGON AND EBERLY (1992). There are three types of univariate interval censorship models:
interval censoring or current status 2" and "Case k" interval censoring,interval CellSC)f]Jlg.
Suppose that YtUllCtiorlS on
data observed are:
The nonparametric maximum likelihood estimation 1" and 2" inf./3rv"J
censored data are well summarized in GROENEBOOM AND VVELLNER (1992), GROENEBOOM (1996),HUANG AND WELLNER (1997). WELLNER (1995) studied the NPMLE of the intervalcensoring in which each subject has exactly k examination times.
To compute the NPMLE, TURNBULL (1976) derived self-consistency equations for a very generalcensoring scheme and proposed solving the equations by the EM algorithm. GROENEBOOM (1991)developed an iterative convex minorant algorithm (ICM). The ICM algorithm is considerably fasterthan the EM algorithm especially when the sample size is large; see GROENEBOOM AND WELLNER
(1992). ARAGON AND EBERLY (1992) and Jongbloed (1995, 1998) proposed modifications of theICM algorithm, and Jongbloed (1995, 1998) showed that his modified algorithm always converges.
Very often in clinical trials, each patient has several follow-ups and the number of followups differs from patient to patient. This motivates the study of the following model. Let TTk,j, j 0,1, ... ,k, k + 1, k = 1, ... , be a triangular array of "potential observation times" withTk,o 0 and Tk,k+I ::::: +00, and let K, the number of observation times, be an integer-valuedrandom variable such that Y and (K, are independent. \Vhat we can observe is a vector X =(2.l.K' TK, K), with possible value x = (Ok, tk, k), where Tk is the kth row of the triangular array T,2.l.k (~k,ll' .. ,2.l.k,k) with 2.l.k,.f = 1(Tk.j_l,Tk,j](Y)' j 1,2, .. , , k + 1. Suppose we observe n i.i.d., . , ex· X' v v rh X· ~ (,A (i) 'T(i) K(i)).. 1 2 'H (}.r(i) 'T,(i) K) ..copleso~ . 1,..i1.2, ... ,An'" ere ~- u
K(i)1 K(i)1 ,Z , , ... ,n. ere ,_, ~ ,z
1,2, ... , are the underling i.i.d. copies of (Y, T, K). SCHICK AND Yu (2000) referred to this model asthe "Mixed Case" interval censoring model. They proved strong consistency in the L 1(fl)-topologyof the NPMLE for a measure fl which is derived from the distribution of observation times.
VAN DER VAART AND WELLNER (2000) gave a different formulation of "Mixed Case" intervalcensoring. They noted that conditional on K and TK , the vector has a multinomial distribution:
where
P(Khas density gk, andmoment that the distribution Gk of (TKIKX is
Suppose forThen the de11sil:y
.1)
with respect to the dominating measure v which is determined by the joint distribution (K,Thus the log-likelihood function for F of b , ... ,Xn is given by
(1.2)1 Ki+l
= - '" '" L::.K.Jlog(F(TK(i)n ~ ~.. . ..i=1 j=1
whereKi+1
mF(X) = L L::.Ki,jlog(F(T~;j=1
K+1
F(T~;,j_1)) L 6.Ki,jlog(6.F(T~;,j))'j=1
VAN DER VAART AND WELLNER (2000) proved the consistency of the NPMLE of the "MixedCase" interval censoring in Hellinger distance, and also recovered the results of SCHICK AND Yu(2000) by using preservation theorems for Glivenko-Cantelli classes. Here, we will use the aboveformulation of the "Mixed Case" interval censoring problem. In Section 2, we give a characterizationof the NPMLE. We then formulate the ICM algorithm for computing the NPMLE. In Section 3, westate the main asymptotic properties of the NPMLE: consistency, global rates of convergence withand without a separation condition, an asymptotic minimax lower bound and pointwise asymptoticdistribution. The results in Section 3 are proved in Section 5 by using modern empirical processtheroy. There are other two estimators for the univariate "mixed case" interval censored data thatare based on the non-homogeneous Poisson process model: nonparametric maximum likelihoodestimator and nonparametric maximum pseudo-likelihood estimator (NPMPLE), see WELLNER ANDZHANG (2000). We denote these two estimators as NPMLEwZ and NPMPLEwz . In Section 4,we present simulation studies to compare the asymptotic relative efficiency of the above threeestimators.
2 Characterization and Computation of the NPMLE
Let tl ::; t2 ::; ... ::; t m denote the ordered distinct observation time points in the set of allobservation time points:
Define the rank function R:
{TKi,j, j = 1, ... , Ki,i = 1, ... , n}.
,j, J 1, ... , Ki, 1, ... , --t {I, 2, ... , such that
= s. if s 1, ... ,
<rewritten as:
F and
the same reason as explained in 2" interval m GROENEBOOM
WELLNER 1992), we can take tl to be an observation point TKi,j such 1,
and to be an observation point such that l(TK (Y'i) 1, Note that n is a convex,In is a concave function due to the linear combination the concave function "log". Thus,the characterization of the NPMLE Fn is given by the following slightly modified version of theFenchel duality theorem in Lemma 3.1 of WELLNER AND ZHAN (1997):
Theorem 2.1 Let ¢ : R n -+ R U { -oo} be a continuous concave function. Let K U R n be a convexcone, and let Ko K n (R). Suppose Ko is nonempty and ¢ is differentiable on Ko. Theni;. E Ko satisfies
if and only if(2.1)
and(2.2)
for all ;f E K.
Note that if;f i;.+dj, where 1j
v¢(i;.))
(i;., ¢(i;.)) 0,
v¢ (i;.)) ::; °(0, ... ,0,1, ... ,1), E > 0, then'-..-' '-..-'
m-j j
0+
Thus, (2.2) becomes
v¢(i;.))
Tn
I: ::; 0,
m
EI:l=m-j+l
::; o.
'i = 1,2, ... , m.this case, any E n, can be written as
where
Ki
Lj=l
and6K(k) = { 1, if R(TK.. i,J) = k,
"J 0, otherwIse.
Then, by applying (2.1) and (2.3), we have the following characterization of the NPMLE Fn :
Theorem 2.2 Let T(l) correspond to an observation time TKi,j such that £lKi,j = 1, and let T(m)be the largest ordered observation times such that the corresponding £lKi,j = 0. Then the unique
NPMLE maximizes In(FIX) over all F E :F if and only if
(2.5)
and
(2.6)
Denote
m n
L L li,k(FnIX) ~ 0,k=l i=l
for l = 1,2, ... ,m,
where
the G(F,') and V(F, processes
== 0,
G(F, =
V(F,O) =
V(F, =
p p
I) -lkk(F1X)) = L L .,.",,,,,."'-,=-=-;k=l k=l i=10,
P
L(lk(FIX) + F(tk)(-lkk(FjX)))k=1
p n
L L [li,k F(tk) li.kk(FIX)] ,k=1 i=1
where p 1,2, ... , m. Since
G(F,pIX)
V(F,pIX) =
p
Lk=1
by using Theorem 4.3 of WELLNER AND ZHAN (1997), we can also characterize the NPMLE Fn asthe slope of the convex minorant of a self-induced cumulative diagram:
Theorem 2.3 Let 1(1) correspond to an observation time 1Ki ,j such that D.K;,j 1, and let
be the largest ordered observation times such that the corresponding D.Ki,j = O. Then is the
NPMLE of Fo if and only if is the left derivative of the convex minorant of the cumulative sumdiagram consisting the following points
Po pI, ... ,m.
convex minorant the cumulative Slun u.W'OL'UH can be
and Proposition 1.4 of GROENEBOOM AND WELLNER 1992). Thus, the computation of the NPMLEof "Mixed Case" interval censoring can be reduced to the "Case 2" interval censoring as notedHUANG AND WELLNER and VAN DER VAART AND WELLNER
The algorithm has been applied in many problems: see GROENEBOOM AND WELLNER (1992),Jongbloed (1995, 1998), WELLNER AND ZHAN (1997), WELLNER AND ZHANG (2000). To prove theglobal convergence, JONGBLOED (1995) designed a modified iterative convex minorant algorithm(MICM) by inserting a binary line search procedure. In this case, the NPMLE of the "MixedCase" interval censoring can be calculated by the MICM algorithm as for the "Case 2" intervalcensoring.
3 Asymptotic Properties of the NPMLE: Results
In this section, we will study the asymptotic properties of the NPMLE: consistency, global rate ofconvergence in Hellinger distance, a local asymptotic minimax lower bound, and pointwise asymptotic distribution of the "toy estimator". The "toy estimator", introduced by GROENEBOOM AND
WELLNER (1992), is obtained by taking one step in the iterative convex minorant algorithm startingwith the real underlying distribution function Fa. It is conjectured that this toy estimator has thesame asymptotic distribution as the NPMLE. The following regularity conditions are needed:
A. EK < 00.
B. Separation condition: there exists a 8 > 0 such that P(TK,j - TK,j-l > 8)J 1,2, ... , K, K = 1,2, ....
1, for every
C. For 0 tk,o < tk,l < ... < tk,k < tk,k+1 == 00, there exists a 77 > 0 such that the underlyingdistribution function satisfies
for all j 1, ... ,k + 1.
D. 1 be the density of the joint distribution Q(tk,j, tk,j-lIK = ,and let be thedensity of with respect to Lebesgue measure on R. Suppose that there exist Co > 0, CI > 0such that
< for t, 8 = 1 2, ... , k 1, ,2, ... ,
x.
1
F. Fix to E . Then there is a neighborhood to such that distribution ofQk,j of is differentiable, and IS cOIltUlUC'US, positive and uniformly boundedallj 1,2, ... ,k,k=1,2, ....
is differentiable, its density is continuous and strictly positive.G. In a neighborhood of
H. In a neighborhood of to, is continuous and strictly positive, where is defined as
(3.7) a(t)
3.1 Consistency
VAN DER VAART AND WELLNER (2000) have proved the consistency of the NPMLE F'n by usingpreservation theorems for Glivenko-Cantelli classes. Here, we give another approach to the proof.Then, following this approach, we continue to study the rate of convergence of the NPMLE.
In the probability space (n, A, P), where n is {O, 1}k+l x Rt x {k}; A is a a-field generatedby Akl x Bk X Ak3' Akl is the class of all subsets of {eJ (0, ... ,~, 0, ... ,0) : j 1, ... ,k + I},
j-th
Bk is a a-field generated by the set of all k-dimensional rectangles on Rt, A k3 = a {0, {k } }; PF isa probability measure with density defined in (1.1) and dominating measure v defined as
(3.8) v({ej} x Bk x {k}) counting measure on {ej}
xP(K = k) x P(T E BklK k).
Q(tk,b"" IKmarginal distribution of Tk,j,Hellinger distance is
be the conditional distribution function of . Let Qk,j denote thefor j = 1,2, ... , k, k 1,2, ... , conditional on K = k. Then the
=
that
elC
~P(Kk=l
= ~P(Kk=l00
= ~P(Kk=l
~Jj=l
J{~ -F(tk'H)]} dq(hlK
k) J[F(tk,k+l) - F(tk,O)] dQ(hi K k)
= ~P(K k) = 1.k=l
For fixed Fo E F and Pp as defined in (1.1), let
(3.9)
(3.10)
(3.11)
(3.12)
P
mp
{pp: F E F},
pp PPo 2pp _ 1.
pp +PPo PP +PPo{mp: F E F},
- {mp - mpo : h(PP,PFb) < 5, FE F}.
Then P is convex: i.e. for all pp, PPo E P, and for 0 E [0,1], opp + (1 0) PPo = Poe Po E P.As shown by VAN DER VAART AND WELLNER (2000), page 123, the Hellinger distance h(PFn,PPo)is less than or equal to PIIM for a density pp with respect to a dominating measure v. Toshow that h(po ,PF'n) -7nq 0, we only need to prove that the class A1 is a Glivenko-Cantelli classby showing t-h'a:t it~ u bra~k~ting numbers are fi~ite for each E > 0, see Theorem 2.4.1 in VAN DER
VAART AND WELLNER (1996). It follows from the following Lemma 3.1 that the bracketing number
(E, A1, Lr(P)) is bounded by (E, P, Lr(Qa)), where dQa for (J > 0.
Lemma 3.1 For any integer r 2 1, let
sup {(J 2 0: J{pp+
dv S; Er
} ,
EP}.
dQ (J = --':::":-t--av
For univariateclass P defined inEK < 00.
the two lemmas show that theand the class ./'-"1 defined in (3.11) are both Glivenko-Cantelli classes under
Lemma 3.2
Lemma 3.3
If EK < 00, P defined in 9) is a Glivenko-Cantelli class.
If EK < 00, A1 defined in 11) is a Glivenko-Cantelli class.
Thus, we have the following theroem about the consistency of the NPMLE with univariate"mixed case" interval censored data.
Theorem 3.1 Under EK < 00, the NPMLE of the univariate "mixed case" interval censored datais consistent in Hell'inger distance.
Consistence of the NPMLE in L1-norm can be derived from the Theorem 3.1; see VAN DER
VAART AND WELLNER (2000). Under additional hypotheses, SCHICK AND Yu (2000) proved that Fn
converges pointwise and even uniformly.
3.2 A Local Asymptotic Minimax Lower Bound
Let Q be a set of probability densities on a measurable space (0, A) with respect to a a-finitedominating measure p, T be a real-valued functional on Q, and let Tn, n ;:: 1, be a sequence ofestimators of T g based on samples of size n from a density g known to be contained in Q. Thenlinimax risk of the estimator Tn, vvhich measures the difficulty of the problem of estimating Tg,is defined as
maxgE9
where 1 is an increasing convex loss function on with l(O) O. n --7 the asymptotic lower bound for the minimax risk can be derived by the following inequality in Lemma 4.1,GROENEBOOM 1996):
The local rates of convergence for 1" and "Case 2" of univariate censoredhave been well studied: see GROENEBOOM (1996). Here we follow the same approach to extendthe minimax results to univariate interval-censored data. Consider the followingperturbations Fn of the underlying distribution function Fo:
(3.14){
Fo(t) + f) fo(to) (t - to +(t) = Fo(t) f) (to) (to + cn
Fo(t),
if t E [toif t E to +otherwise,
, to),
where 0 < f) < 1 is a constant, c is a constant to be determined by Fo and Q. Then the densityfunction f n of Fn is
Note that
{
fo(t) + f) fo(to),fn(t) fo(t) - f) fo(to),
fo(t),
ift E [t cn- 1/ 3 t )o ,0,if t E [to, to + cn- 1/ 3J,otherwise.
{ fn(t)dt = ( fo(t)dt + f) fo(to) ( {to dt (toto+cn-1/3 dt) = 1,i R+ i R+ ito-en-1/3 it,
and for all t E [to, to + cn- 1/ 3J,
fn(t) = fo(t) - f) fo(to) = (1 - f))fo(to) + O(n-1/3
) > 0,
for 0 < f) < 1, and large n. Thus, (3.14) is a sequence of univariate distribution functions.For "Mixed Case" univariate interval-censored data, an asymptotic minimax result is given by
the following theorem.
Theorem 3.2 Let Fo(t) be the under-lying distr-ibution function with density , let Fn be thesequence of per-tur-bations of Fo by (3.14), qkO 0, qk,k+l == 0, qk,j(t) be the dens'ity of thedistr-ibution function Q(tk,jIK = k), and let be the joint density of Q(tk,j, IK =for- j 1, ... , k, k 1, .... Suppose that condition B holds and
00
LP(K + < 00.
{ }
Co
00
LP(Kk=l
,is a constant ael)enaw"q on (), and
Note that in the cases of P(K = 1) and P(K 2), the above a(to) reduces to the known lowerbound results for univariate current status data and univariate interval censored data: case 2; seeGROENEBOOM (1996). In the case of bivariate interval censored data, the local rate of convergenceis also shown to be n l/3 ; see SONG (2001).
3.3 Global Rate of Convergence in Hellinger Distance
We will apply empirical processes theory to study the rate of convergence of the NPMLE of withunivariate "mixed case" interval censored data. Consider a deterministic function M : e i-7 R,and stochastic processes indexed by a semimetric space e. Let ()o be a point of maximumof the map () i-7 M(8). Let an be estimators that (nearly) maximize the maps () (8). Anupper bound for the rate of convergence of an can be obtained from the continuity modulus of thedifference see Theorem 3.2.5 in VAN DER VAART AND WELLNER (1996). In the case ofi.i.d. data, (8) = (8) and M(8) P(8), the centered process J1l(Mn - = (mo) is anempirical process at mo. Define
= {mo - moo: d(8,()o) < 6}.
Then Theorem 3.2.5 in VAN DER VAART AND WELLNER (1996) leads to the following corollary,
Corollary 3.1 In the i.i.d. case, assume that, for every () in a neighborhood of 00 ,
P(mo );S
C~ < 2a Jurletlion some
key step to apply Corollary 3.1 is to derive sharp bounds on the continuity modulus of theempirical process. Define the uniform entropy integral for the class by
J(ry, sup (7P Jo
and the bracketing integral
(ry,
where iUS is an envelope function of the class . Then the bounds can be obtained by the twomaximal inequalities given by Theorem 2.14.1 and 2.14.2 in VAN DER VAART AND WELLNER (1996):
Ep :s J(l, MIs) (p* lVIn1
/2
,
EpIIGnIII~ ~ J[](1,MIs,L2 (P)) (P*Ml) 1/2 .
An alternative is to use Lemma 3.4.2 in VAN DER VAART AND WELLNER (1996),
where each element in is uniformly bounded by },;I. We will apply Corollary 3.1 to study therate of convergence of the NPMLE with the univariate "mixed case" interval censored data. Therates are obtained with and without a separation condition. We first obtain the entropy withbracketing for the class Ai with density defined in (1.1).
Lemma 3.4 Under conditions .4, Band C,
Under the separation condition, the following theorem gives the global rate of convergence inHellinger distance for the NPMLE of univariate "mixed case" interval censored data.
Theorem 3.3 Under conditions .4, Band C, h(PF'n,PFO)
Without the sej:)aration condition B, the global rate of convergence in J.J.C;ULU.i".'"J. distance of theNPMLE with univariate "mixed case" interval data is
Theorem 3.4 CorWZic201"/'S A and D,
3.4 Pointwise Asymptotic Distribution
GROENEBOOM AND WELLNER (1992) conjectured that the estimator" has the same asymptoticdistribution as the NPMLE for the univariate interval censored data: "case This cOJojectllfeis called the "working hypothesis" and supported by numerical results. The conjecture has beenproved under a separation condition for the univariate interval censored data: "case 2", see GROENE
BOOM (1996) , pages 138-159. Here we will study the pointwise asymptotic distribution of the "toyversion" NPMLE for the univariate "Mixed Case" interval censorship model.
Suppose the marginal distribution Q(tk,llK = k) has density Qk,1 (tk,1) with respect to Lebesguemeasure on R; the marginal distribution Q(tk,kIK k) has density qk,k(tk,k) with respect toLebesgue measure on R; joint distribution Q(tk,j, tk,j+1IK = k) has density Qk,j,j+1(tk,j, tk,j+1)with respect to Lebesgue measure on R2 for j 1,2, ... ,k -1, k 2,3, .... The following theoremgives the pointwise asymptotic distribution of the NPMLE for the univariate "mixed case" intervalcensored data.
Theorem 3.5 Under conditions B, E, F, G, H, then the toy estimator Fr~1) satisfies
(3.15)
where Z (t) is a, standard two-sided Brownian motion on R starting .from 0, and a(t) is defined in(3.7).
To get a version of Theorem 3.5 for the NPMLE, we first need to show that, for each lVI > 0,
(3.16)
and
sup 'FA (\ D ( ) I 0 ( -1/3I n t) - L' 0 to = p n ' ' ),
(3.17)
under some regularity conditions. Then (3.16) (3.17) will be used with some empirical processesarguments to show that the difference between the process
process
{ + }- { + }
in t of the process (3. and the distribution of the process t -+ 11,1/3 + tn- isdetermined by the local minima in t of the process (3.19). For details on the "switching relation",see pages 91-95 in GROENEBOOM AND WELLNER (1992) and examples 3.2.14 and 3.2.15 in VAN DER
VAART AND WELLNER (1996). The pointwise asymptotic distribution of (to) - Fa }followsfrom the Theorem 3.5. One difficulty here is that considerable work is needed to prove that (3.16)and (3.17) hold.
4 Efficiency of the NPMLE
Let Fn denote the NPMLE in the current model. Let An and A~s denote the NPMLEwZ andNPMPLEwZ in the non-homogeneous Poisson process model, see WELLNER AND ZHANG (2000).Note that the one-jump counting process N(t) = 1[Y::;t] has mean function A(t) = E{N(t)} =P(Y :s; t) F(t); see example 4.1, page 792, WELLNER AND ZHANG (2000). Based on the studyof the pointwise asymptotic distribution of the Fn , An and A~s, see Theorem 3.5 in Section 3,Theorems 4.3 and 4.4 in WELLNER AND ZHANG (2000), we suppose that the estimators Fn , An andA~s have the same asymptotic distribution up to positive constants L 1, L 2 and L3:
and
the unit
UIC'LLJ.UU.LJ~Jllwith
113 A • 2 i311, i (Apsn(to) - Ao(to)) -+d L 3/ X,
where X is a known random variable. Note that Var(Fn) ~ 11,-2/3Li/3 Var(x), Var(An) ~ 11,-2/3L~/3
Var(x), and Var(A~S) ~ 11,-2/3L;/3 Var(x). To study the asymptotic relative efficiency of twoestimators, we ask the two estimators to have the same variance, asymptotically. Thus, we can plotthe 3/2 power of the ratio of sample variance of the two estimators to obtain an approximation tothe asymptotic relative efficiency. WELLNER AND ZHANG (2000) showed the asymptotic efficiencygain of An over in their Figures 3 and 4. In this simulation study, we study the efficiency gainof the over An by plotting (Var(Fn)/Var(An))3/2.
Random samples of univariate "mixed case" interval censored data Xii = 1, , n, were as follows: K i E {I, 2, 3, 4, 5, 6, 7, with
k = 1, , 8. are ordered sequence K i random variables gelleratE:d
time
.~.~ .-"
:~.,...
3.02.52.01.51.0
~
:>,<X><.>
55 0
~<D
Q) 0.2:<6C30:: ....<.> 0'"515.E N:>,
oil 0
<=>0
0.0 0.5
Time
Figure 1: The asymptotic relative efficiency of Fn versus the An. N is the number of simulationruns; n is the number of subjects.
the number of subjects n 1000. The estimated efficiency gain of F'n over An is plotted in Figure1. As shown in Figure 1, the NPMLE Fn is more efficient than the NPMLE An: for most of thosepoints where the Fn and An were estimated, the estimated asymptotic relative efficiency is above40% for both simulation cases. When time is between 0.2 and 1.6, the estimated asymptotic relativeefficiency for the simulation with n = 100 is close to that for the simulation with n = 1000; whenthe time varies from 1.8 to 3.0, the estimated asymptotic relative efficiency for the simulation withn 100 is generally higher than that for the simulation case with n = 1000. Although Theorem3.5 and Theorem 4.4 in WELLNER AND ZHANG (2000) show that the asymptotic relative efficiencyfor the toy estimators of Fn and An is equal to 1 for a one jump process, the log-likelihood functionof (2.7) for a one jump process in WELLNER AND ZHANG (2000) is different from that of (1.2). Thissimulation shows that we need to study asymptotic distributions of the two estimators F'r, and An.
5 Asymptotic Properties of the NPl\1LE: Proof
proofs ofsectioJn, we will give
Proof of Lemma 3.1
as)Tm]ptoitIc properties
tolJowin!! brackets for A1:
::;eCILlon 3.
where , i 1, ... ,I, are for class P. Since is de<::reaslng in t, and
so for all mF E A1, there exists i E {I, ... ,I} such that m~,i :::; mF :::; and<
:::;1.
Thus, +2 -m~1 , and
- m~llp,r {/ ImimWdP}
< {2r[/ PFo l[ppo::;£T]dV] + / Imi - mW l[ppo > £T]dP}
< (2r Er + Er)l/r ,
if II(mi mD1pro>£TIIP,r:::; E. This implies that
Also,
PF',i + PFo
PFo
IPF,i
P~,i +PFo
r
dv
because -;---"'-- :::; 1, and -c;;--...JL-- < 1. Define
Proof of Lemma 3.2 Since F is a class bounded monotone functions: F : Rfor every probability measure Q, by Theorem 2.7.5 in VAN DER VAART AND WELLNER
exists a constant K depending on T 1 here), such
1] thenthere
logNr" l
for E > O. Let I (E,F,LdQ)). Then for the probability measure P(TK,j BIK =PK,j(B), where B E B, there exists a set of brackets:
[F{ , F{ ] ' ... , [Ff, F[ ] ' ... , 1, ... ,I,
such that for any F E F, Ff (t) :s; F(t):S; (t) for all t and somei, and EpK,j IFf (tk,j) - F[ (tk,j) I :s; E.
Since
and
it follows that
Let the corresponding brackets for P be defined by
(5.20)K+1
P~,i II (Ft(tK,j) F[(tK,j-d
(5.21)
Then
! !
K+l
== II (F[j=l
{
dv PFo :s;
K
::; 2 E ('2:: P(Kk=l
if EK < 00. This implies that:
= 2E(K l)E < 00,
logN[ (2EE(K+1),P,L1(P))
Thus, by Theorem 2.4.1 in VAN DER VAART AND WELLNER (1996), P is a Glivenko-Cantelli classunder EK < 00. 0
Remark: From the above proof, we also have that
(5.22)
if EK < 00. Here v is defined in (3.8).
Proof of Lemma 3.3 Let a E/(EK). Then,
«uPlodV < E~! dv ~ E~ {~P(K k) ~! dQ(clK~ k)},..,E
uE(K + 1) = (1 + ,..,lrT) E.
£;1\ \ £;1\ J
Let r = 1 in Lemma 3.1. Then dQer =
Then it follows that
IPF,i
<
dv, so
<
00.
Proof Theorem 3.
Proof of Theorem 3.2 Define
if t E
ift E
otherwise,
Note that
andk+1L Ok,j (Fn(tk,j) - Fn(tk,j-d)j=lk+l k+l
L Ok,j (Fo(tk,j) - FO(tk,j-1)) + (B fo(to)) L Ok,j (Hn(tk,j) - Hn(tk,j-d)j=l j=l
k+1PFo(rl,tJ + (B fo(to)) L Ok,j (Hn(tk,j) - Hn(tk,j-d)·
j=l
Under the separation condition B and for large n, tk,j and tk,j-1 will not be in :In at the sametime. This indicates that
Thus
and
( 1 )2+y'PFo
1 '3+ O(n- 1/. ) 1 l'.t)
4- In ' ,PFo
00
LP(Kk=l
k+l
k) L J(Hn(tk,j) Hnj=l
I:{[ H~. In
iK
}
2
+2
if00
I:: P(K = I::(qk,j(tO) + < 00,
k=1 j=1
The above condition is satisfied if EK < 00 and qk,j is bounded above for all j = 1" , , ,k, k = 1" , , ,The Hellinger distance becomes
H2
(PFn ,PFb) = ~ j(VPFn - VPFo)2dv = 21
j (J:i=p~)2 dv- . PF n ' vP~
-81 j (PF'n -PFo)2 dv + O(n- 1/ 3 );' (PFn PFo)2 dv
PFb
~ j (PFn PfrJ2
dv + o(n- 1 ),
8 PFoNow,
Thus,
Now, by inequality (3.13),
+ ).
infmaxTn
I En,PFo Fo }
1~ 4 IFn(to)
1~ 4 0 c f 0 ( to) (1
1-t -Ocfo(to) e
4
The last expression is maximized by
Fo(to)l{l H 2(PFn' PFo)}2n
1 .4" a(to) (Oio + o(n- 1
as n -t 00.
_ { 2/3 }c = a(to) 02 f6(to)
and the maximum value is
Thus,
~ (20/3) e- 1/ 3 (fo(to)) 1/3
a(to)
liminfn max {En,PFnn-+oo {
f}
1/31 > 0
tJ - Co a(to)
where Co i (20/3) is a constant depending on O. o
Proof of Lemma 3.4 By Theorem 2.7.5 in VAN DER VAART AND WELLNER (1996), for all E > 0and for any probability measure Q, there exists a constant K such that
such < <
,K
To the brackets of collection note that
TlLF
Also,a
-c---,-;::- > 0(x +
for a > 0, and under conditions Band C, there exists a T7 > 0 such that Fo(TK,j) - Fo(TK,j-1) > T7
for all j = 1,2, ... , K, K = 1,2, .... Thus, 0 < flFo,K,j ::; 1, and t:.FK~:~'h,K,j is increasing inflFK,j ;::: O. Choose the brackets of M as
1,K+1
mUX) = 2 L 6K,j ~--"-'----"''-----''-''':'-~--'---j=l
K+l Fr() l( )2 L 6K', ,i TK,j Fi TK,j-1 - 1.
j=l ,J [F[(TK,j) - F;(TK,j-d] + flFo,K,j ,mi(X)
where i = 1, ... , I. Then
~P( ~ r -m~12 PF~1dlJ4 ' 4 J
~ P(K d)~J{-;:------"-~:...,---"-'-:!"--'----
-=-:-----"--'---~---"---'---c==--'----- } 2 flF.u' .k dQ (tk !K =flF.' ' ,J - IO,K,J
00
< LP(K=
2
00
< 2LP(K ~{J
J
2dQChl K
}00
< 2LP(K=k=l
k+1
L(E2 +j=l
(EK + 1) < 00,
if EK < 00. This implies that
Proof of Theorem 3.3 In order to apply Corollary 3.1, we need to use Lemma 3.4.2 in VAN DER
VAART AND WELLNER (1996) to find the upper bound for E*IIGnIIMs' where
A15 {mF : h(PF,PFo) < 5}.
For mF E A15,
J. J( )2PF PFim~PFo dv = 0 PFo dv
PF + PFo
r!PF - PFol. P~o dv'. PF + PFo PF T PFo
< JIPF PFol dv dTv(pj, PFb) ::; V2h(PF,PFo) < J2 5.
Also, allmF E1996),
E*
::; 1. Thus, by Lemma 3.4.2 in VAN DER VAART
(1 + -"-'----:::----==----
WELLNER
Corollary 3.1,
From
<
we have r n ::; . Thus, the rate is o
1>0-1 dv.
·PFo
Proof of Theorem 3.4 We will use Lemma 3.1 to control the bracketing entropy of the classA1. In this case, r = 2, so
1l[pF >0-1- dv.
o . PFo
Let [pL pi] be a pair of bracket for the class P as defined in (5.20), (5.21) . Note that
Define1
>0-1 J'Pf'o
Then
/ II( =
::; {/(F[ dQ(hiK =
Thus,
This implies that
0"1-1- dV}
lPFO
Now for fixed k :::: 1 and 1 ::; j ::; k,
/l[FoCtk,j
Co, c do not depend on j and k by assumption D. Hence we have
/k
L
Take a . Then, if EK < 00,
10gN[ P, < (1 1 1/2.1.)'" - og' (-) .E E
By Lemma 3.1,
We have shown that
It follows that
E*
J[ (<5, lvl8, L 2 (P)) 18 VI + 10gN[](E, ~\18, L 2 (P)) dE
< 18J~10g1/2(~)dE::S<51/210g1/4~.
By Lemma 3.4.2 in VAN DER VAART AND WELLNER (1996),
<' . 1 <51/210g1/41 . 1 100'1/2II .. ;::, <51/2 1 1/4 (1 + ,) ) = <5 1/ 2 1 .1/4 + _o=--,,-M" og <5 yin <52 og <5 <5
By Corollary 3.1,
¢(<5).
Thus, the rate is
Tn ::; 10g-1/6n .
10g-1/6n . This completes the proof. o
Proof of Theorem 3.5 Define the processes
[~(~+~Ki
(~
define the processes:
=
We will use Theorem 2.11.23, see VAN DER VAART AND WELLNER (1996), to show that the process
(5.26) { + (to) }
= { (to + (to) }
+lto+n-1/3t (Fo(s) Fo(to)) dGhO) (s)
to
converges in distribution to the process U(t) defined by
(5.27)
where t E R. First, note that
l/3t]
where
and
(X)K I AL ( UK,j. !1Fo K)'
)=1 '
!1K '. 1 "),)+ . (1!1Fo,K,j+1
),
00
LP(K=k=l
o.
Then
2
<
r}
By assumptions B, F and G, there exist C > 0, C\ > 0 such that FO(tk,j) - FO(tk,j-1) 2:: C- 1, andQ~,j(t) ::; C1, for j 1,2, ... ,k, k 1,2, .... Also by assumption E, we have
co
2Cn1/
3 LP(Kk=l
)
2 K.6.K,j+l . 1
.6.FoK+1 L [,) , )=1
-toi<Bn-
dQk,j
. we show all T/ > 0, O. For all E > O.
<
as n 00.) --+ 0,\2}
and tl > t2. We
K
(X) '"~ t..FoK,j
Then
t..K,j+l )t..FoK,j t..FoK,j+l
1/3P (K ( t..K,j _, t..K,j+1 ) 2
< n ~\t..FOK,j D.FOK,j+1
~ 1 ' -lf3t ,). L [to+n- I/ 3tz<Tg,j"5: to+n I IJ
j=l
00 J[k ( t..K,j t..K,j+l) 2- n
1/3t; P(K = k) ~ t..FoK,j - t..FoK,j+1
K ].L -to"5:n-1/3tJ] dvj=l
k=l
1
t..Fb K,j+l
iK=
Construct a set brackets for class by first partitioning - B, B] as
-B < < ... < 0< < ... < hq = B
such that < E, for 1 :s; i :s; q and q = O(some i E {O, 1, ... ,q} such that the bracket
Then for given t E
is
1, ,there exists
{
K (I _ Lj=l
fn,i - _ 1/6 ,,",Kn L..tJ=l
n 1/6 ,,",K ( fJ.K,;L..tJ=l fJ.FoK,j
_n1/ 6 ,,",K (-:cri-'-'--'-L..tJ=l
As just shown above
fJ.K,j+l ) 1fJ.Fo K,j+l [to<TK,j:StO+n - 1
/ 3hd'
K,j+l 1[to+n- 1/ 3h i <TK,j::;toj'
if i ~ p + 1,
if i :s; p.
P(J~,t f~,t)2 ~nl/3p ( {~ ~K,j+1 )
~Fo K,j ~Fo K,j+I
It follows that
which implies that
Thus,
dE =
5 < 5, E
1,vlK = k)
dv
IK = k)
(1
k-1
+2.:j=l
+
[
K
'f;
{
! (K [( LiK' ) 2 (Li T/L3p 2.: ,J ! _---'H~_
n ' j=l LiFo K,j T LiFo K,j+1
fP(K k)jt ( 1 + 1j=l j=l LiFo K,j LiFo K,j+1
t, P(l( k) {t Fb(:ktl
=
Similarly, if s < t t E
If s x t < 0, dearly = O. Then
) - \ ( ,) -7 -t a to).
P(j(O) j(O)) _ P(/O))P(j(O)) = O.n,t n,s n,t n,s
Thus, by Theorem 2.11.23, see VAN DER VAART AND WELLNER (1996),
in FN[-B, B],
where Zoo [- B, BJ is set of all uniformly bounded real functions on [-B, BJ.By assumptions of B, F, G and H, the expectation of the second term in (5.26) is
dQ(t IK-kl }
in
2t .
Drc)vi(ied that
To
and
that
(h)
Op(l).
we define the following two sto,ch,'tstic processes:
P) [t (D.~Kil] .> - -=-::.~:......j=1 OK t ,]
+ rto+\Fo(s) _ Fo(to))dG~O)Jto
M(h) E [l:O+h(Fo(S) - Fo(to))dG~O)(s)].
It follows that to prove (5.28) is equivalently to prove
n 1/ 3hn = n 1
/ 3argminn Mn (h) Op(1).
Note that ho = 0 is the only location of the minima of M(h). As shown before, we have, usingconditions F, G and H,
M(h) M(ho)r rto+h ]
E lito (Fo(s) Fo(to) )dG~O) (.s)
lto+h
(Fo(s) - Fo(to))a(s)d.s.to
~ (to)a(to)h2 o(h2),
as h -+ O. The difference between the process (h) and the process is
(h)
ro+h
Itod
+n [t ( CiKi,J 2 +--=~~j=1 (CiFOKi,j) (CiFoKi,j+l)
. (1[TKi,j9o+hj l[TKi,j (Fo(TKi,j) - Fo(tO))]
Combining this with the fact that (ho) M(ho) = 0 yields
(ho)1(h) -
[t CiFo K ' - --'-"-'~)=1 to)
Gn [t (( CiKi,j 2 + -,-----"=------:;:
j=1 CiFOKi,j) (CiFoKi ,j+1
vnl
Define a class :F~) as
) : ::;D~.J
The envelope function of the class :F~) can be chosen as
-=~L:!:.!:_I
CiRo' K' '-'-1 1t,J I
r2. we
E
Following the same calic:ul:a,tlC)ll of E
!{t 2 K
} NOd". I:1[
I:P(K It{ 1=
k=l
K
} dQ(!kI K'I:1[ k)j=l
00 k
< 2C I: P(K k) k JI: 1[ dQk,jk=l j=l
= 2cf P(K = k) k [t rt°+D
Q~,j(s) dS]k=l j=l Jto-D
< 4C C1D (EK2) < 00,
by conditions B, F and G. Similarly, following the same argument that gave
we obtain
It follows that
dE = 0(1).
Thus we using Theorem 2.14.2 in VAN DER VAART AND WELLNER (1996),
<~ o
{a as
Let be envelope function of the
- Fa(Fo--~-+--~-'---
, 2Following the same calculation of E[FJ})] < 00 and arguments that yield
we obtain,
[(oJ] 2E H V < 00,
by conditions of B, F, G, and
N[] (EI1F1~~llp,2,F~,~,L2(P)) 0 (E12 ) .
It follows from Theoren1 2.14.2 in VAN DER VAART AND WELLNER (1996) again, we have
All these show that the function <Pn(S) in the Theorem 3.2.5 of VAN DER VAART AND WELLNER
(1996) can be taken as <Pn(S) = 15 1/ 2 that yields Tn n 1/ 3 in the Theorem 3.2.5 of VAN DER VAART
AND WELLNER (1996). Hence (5.28) is proved.By using Exercise 3.2.5 of VAN DER VAART AND WELLNER (1996) and symmetry of Brownian
motion,
d
d
(2f6(t:)a(to) )
( 2f6(t:)a(toJ
Thus, following the same argument as in GROENEBOOM AND WELLNER (1992), the
satisfiesestimator
>
>
. 2 argma:x.f
>
is
This completes the proof.
·2 argma:Kf
o
Acknowledgments
This paper is part of author's Ph.D. dissertation under supervision of Professor Jon A. Wellner.Part of the writing and computation were carried out while the anthor was employed by theBoeing Company. This research was partially supported by National Science Foundation grantDMS-9971951.
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Mathematics & Computing Technology, The Boeing Company, P.O. Box 3707, MS 7L-22, Seattle,Washington 98124-2207, U. S. A.E-mail: [email protected]
