Conditional Quantile Estimation with Truncated,Censored and Dependent Data?

Hanying LIANGDeli LITianxuan MIAO

1 Introduction

In medical follow-up or engineering life testing studies,one may not be able to observe the variable of interest,referred to hereafter as the lifetime.In this paper,we focus on the lifetime data with multivariate covariates which are subject to both left truncation and right censorship.Let(X,Y,T,W)be a random vector,whereYis the random lifetime with the distribution function(df)F,Tis the random left truncation time with the dfL,Wdenotes the random right censoring time with dfGandXis anvalued random vector of covariates related withY.Assume thatXadmits the dfM(·)and densitym(·).

In the random left truncation and the right censoring(LTRC)model,one observes(X,Z,T,δ)ifZ≥T,whereZ=min(Y,W)andδ=I(Y≤W);whenZ0.If(Xi,Zi,Ti,δi),fori=1,2,···,n,is a stationary random sample from(X,Z,T,δ)which one observes,then(Ti≤Zi,?i).Without loss of generality,we assume thatY,TandWare nonnegative random variables as are usual in survival analysis.Following the idea of Iglesias-Pérez and González-Manteiga[13],we define a generalized product-limit estimator(GPLE)of the conditional distribution functionF(y|x)ofY,givenfor the LTRC data by

whereBni(x)K(·)denotes a kernel function onand 0

One characteristic of the conditional distribution functionF(y|x)that is of interest is the conditional quantile function.It plays an important role in various statistical applications,especially in data modeling,reliability,and medical studies.Letξp(x)=infforp∈(0,1)be the conditional quantile function ofF(y|x).We focus here on estimatingbased on the LTRC data.A natural estimator ofis given by(x)=infIglesias-Pérez[14] first derived an almost sure representation and the asymptotic normality ofunder i.i.d.assumptions and the cased=1.

Asymptotic properties for di ff erent quantile estimators with censored and/or truncated data have been studied by many authors.In the absence of covariables,representations of the product-limit quantile estimator were obtained by Lo and Singh[21]for censored data,by Gürler et al.[10]for truncated data;asymptotic normality and a Berry-Esseen-type bound for the kernel quantile estimator were derived by Zhou et al.[31]for jointly censored and truncated data.In the presence of covariables,we cite the representations derived by Dabrowska[6]and Van Keilegom and Veraverbeke[28]for conditional quantile estimators with censored data,the strong uniform convergence with rate for a kernel estimator of the conditional quantile established by Ould-Sa?d[23]for censored data,and the asymptotic properties of the kernel conditional quantile estimator for the left-truncated model studied by Lemdani et al.[16].In all of these papers,it is assumed that the observations are independent.

However,the dependent data scenario is an important one in a number of applications with survival data.When sampling clusters of individuals(family members,or repeated measurements on the same individual,for example),lifetimes within clusters are typically correlated(see[3,15]).There has been some literature devoted to the study of the conditional quantile estimation under dependence.To mention some examples,Cai[4]investigated the asymptotic normality of a weighted Nadaraya-Watson conditional quantile estimator for theα-mixing time series.Honda[12]dealt withα-mixing processes and proved the uniform convergence and asymptotic normality of an estimate ofξp(x)for the cased=1 using the local polynomial fitting method.Ferraty et al.[8]considered quantile regression under dependence when the conditioning variable is in finite dimensional.A nice extension of the conditional quantile process theory to set-indexed processes under strong mixing was establish in[26].Ould-Sa?d et al.[24]recently discussed strong uniform convergence with rate of the kernel conditional quantile estimator with left-truncated and dependent data.Liang and de Un?a-lvarez[18]proved the strong uniform convergence and asymptotic normality for the kernel estimator of the conditional quantile under censored and dependent assumptions.The asymptotic normality of the conditional quantile estimator with auxiliary information for left-truncated and dependent data was discussed by Liang and de U?a-álvarez[19].However,to the best of our knowledge,the asymptotic properties of the conditional quantile estimator with dependent data for the LTRC model have not yet been investigated.

In this paper,we study the strong convergence with rate,strong representation as well as asymptotic normality of the conditional quantile estimatorwhen the observations with multivariate covariates form a stationaryα-mixing sequence.Also,a Berry-Esseen-type bound for the estimator is established;this result is new,even for independent data.The finite sample behavior of the estimator is also investigated via simulations.

In the sequel,{(Xi,Zi,Ti,δi),1≤i≤n}is assumed to be a stationaryα-mixing sequence of random vectors.Recall that a sequence{ζk,k≥1}is said to beα-mixing if theα-mixing coefficient

converges to zero asn→∞,wheredenotes theσ-algebra generated bywithl≤m.Among various mixing conditions used in the literature,αmixing is reasonably weak and known to be ful filled for many stochastic processes including many time series models.Withers[29]derived conditions under which a linear process isαmixing.In fact,under very mild assumptions,linear autoregressive and more generally bilinear time series models are strongly mixing with mixing coefficients decaying exponentially,i.e.,α(k)=O(ρk)for some 0<ρ<1.See[7,p.99],for more details.We mention thatα-mixing has been used in applications with clustered survival data;see,for instance,Cai and Kim[5].

In the sequel,for any dfQ(y)=P(η≤y),we denote its density function byq(y),and the left and right support endpoints byaQ=inf{y:Q(y)>0}andbQ=sup{y:Q(y)<1},respectively.Forxde fineθ(x)=P(T≤Z|X=x),

Also,we de fineQ(y|x)=P(η≤y|X=x)andQ?(y)=P(η≤y|T≤Z),while their density functions are denoted byq(y|x)andq?(y),respectively.ThusM?(x)=P(X≤x|T≤Z),and its density function ism?(x).

Remark 1.1It is easy to verify thatm?(x)=θ?1θ(x)m(x).Assuming thatY,TandWare conditionally independent atX=x,andF(·|x)andG(·|x)are continuous,thenC(y|x)=θ?1(x)L(y|x)(1?G(y|x))(1?F(y|x))=θ?1(x)L(y|x)(1?H(y|x)),and=which gives=θ?1(x)L(y|x)(1?G(y|x))f(y|x).

De fine estimators of(·|x),C(·|x)andm?(x)respectively as follows:

and

The rest of this paper is organized as follows.The main results are described in Section 2.A simulation study is presented in Section 3.All proofs are given in Section 4.Some preliminary lemmas,which are used in the proofs of the main results,are collected in Appendix.

2 The Main Results

Throughout this paper,x=(x1,···,xd)For(i,j)=(i1,···,id,j)putf(i,j)(y|x):=LetC,C1,···andc0,c1,···denote generic finite positive constants,whose values may change from line to line,and let Φ(u)stand for the standard normal distribution function and[t]be the integer part oft.The notationAn=O(Bn)meansandU(x)represents a neighborhood ofx.LetIbe a compact set ofwhich is included inD={x|m(x)>0,θ(x)>0}.Sete=(e1,···,ed)for smallei>0,andIe={x±e,x∈I}with{m(x),θ(x)}≥δ0>0.

Throughout this paper,we assume thatα(k)=O(k?λ)for someλ>0.We first list the following basic assumptions:

(A1)(i)K(·)is a Lipschitz-continuous density function with compact support on

(ii)K(x)dx=0 for non-negative integersi1,···,idwithi1+···+id≤r0?1.

(A1’)(i)K(·)is a bounded density function with compact support on

(ii)K(x)dx=0 for non-negative integersi1,···,idwithi1+···+id≤r0?1.

(A2)(i)Y,TandWare conditionally independent atX=sfors

(ii)τ1andτ2are two real numbers such thataL(·|x)<τ1≤τ2

(A2’)(i)Y,TandWare conditionally independent atX=sfors∈U(x);

(ii)τ1andτ2are two real numbers such thataL(·|x)<τ1≤τ2

(A3)The firstr0partial derivatives of functionsθ(s)andm(s)are bounded fors∈Ie,and the firstr0partial derivatives with respect tosof functionsL(y|s),G(y|s),F(y|s),l(y|s),g(y|s)andf(y|s)are bounded for(s,y)∈Ie×

(A3’)The firstr0partial derivatives of functionsθ(s)andm(s)are bounded fors∈U(x),and the firstr0partial derivatives with respect tosof functionsL(y|s),G(y|s),F(y|s),l(y|s),g(y|s)andf(y|s)are bounded for(s,y)∈U(x)×

(A4)For all integersj≥1,the joint conditional density(·,·)ofX1andXj+1exists onand satis fies(s1,s2)≤C1for(s1,s2)∈Ie×Ie.

(A4’)For all integersj≥1,the joint conditional density(·,·)ofX1andXj+1exists onand satis fiesfor(s1,s2)∈U(x)×U(x).

(B2)For all integersj≥1,the joint conditional density(·,·,·,·)ofexists on×[0,1]×[0,1]and satisfies(s1,s2,y1,y2)≤C3for(s1,s2,y1,y2)∈U(x)×U(x)×[0,1]×[0,1].

Remark 2.1(a)(i)and(ii)in(B3)imply(i)and(ii)in(A5’),respectively.

(b)Similar conditions as(A1)–(A3),(A1’)–(A3’)and(B1)have been used commonly in the literature,see,e.g.,Iglesias-Pérez and González-Manteiga[13]in the casesd=1 andr0=2.The role of conditionin(A2)and(A2’)is to avoid the problem that the conditional functionC(y|x)may vanish.The conditions(A3)and(A3’)allow us to apply Taylor expansions in the proofs to determine the order of convergence of the estimators.Conditions(A4),(A4’)and(B2)are mainly technical,which are employed to simplify the calculations of covariances in the proofs,and are otherwise redundant for the independent setting.

(c)Assumptions(A5),(A5’)and(B3)imply restrictions on the degree of dependence of the observable sequence;as we discuss now,the message under these assumptions is that one must prevent strongly dependent data.Indeed,all these conditions are satis fied by appropriately choosing the bandwidthhnwhenλis large enough.Note that,if the exponential decayα(k)=O(ρk)for some 0<ρ<1,which has been used by some authors(see[7]),we replaceα(k)=O(k?λ),and thenλcan be arbitrarily large.

In order to give the strong convergence with rates of(x),we need the following additional assumptions:

(D1)For each fixedp∈ (0,1),the function(x)satis fies that for any>0 and any functionηp(x),there existsβ>0 such thatimplies that

(D1’)For each fixedp∈(0,1),the functionξp(x)satis fies that for any0 and any functionηp(x),there existsβ>0 such that|ξp(x)?ηp(x)|≥implies that|F(ξp(x)|x)?F(ηp(x)|x)|≥β.

(D2)There existsγ1>0 such that

(D2’)There existsγ1>0 such that

Theorem 2.1Let α(n)=O(n?λ)for some λ>2.

(a)Let0

(b)Letx∈D and0

In order to formulate the strong representation and asymptotic normality of(x),we need to impose the following additional assumptions:

(E1)f(0,1)(y|x)is bounded fory∈[τ1,τ2].

(E2)The sequenceα(n)satis fies for positive integersq:=qnthatq=and

(E3)→0.

Theorem 2.2Set ξ(Z,T,δ,y,x)=(t|x).Let α(n)=O(n?λ)for some λ>6,letx∈D and0

where for i=1,2,a.s.when(B3)(i)holds;|Rni(ξp(x)|x)|=Opwhen(i)and(B3)(ii)hold.

Theorem 2.3Let α(n)=O(n?λ)for some λ>6,letx∈D and0

where

In order to give a Berry-Esseen-type bound for(x)which will assess the quality of the normal approximation in Theorem 2.3,we need the following additional assumption.

(Q)p:=pnandq:=qnare positive integers such thatp+q≤n,0 andqp?1→0.

Theorem 2.4Let α(n)=O(n?λ)for some λ>withand letx∈D and0

Remark 2.2The assumptionsγin→0(i=1,···,5)in Theorem 2.4 can be satis fied by appropriate choice ofhn,pandq,whenλis large enough(note that if we replaceα(n)=O(n?λ)by the exponential decayα(n)=O(ρn)for some 0<ρ<1,thenλcan be arbitrarily large).In particular,choosingp=[ns]andq=for someandmaxthenγin→0(i=1,···,5),qp?1→0,pn?1→0,and(B3)(i)holds.

3 Simulation Study

In this section,we investigate with simulated data the finite sample performance of the proposed estimator(x)withp=0.5 in the cased=1.In particular,we calculate the mean squared error(MSE),plot the Boxplots of the estimatoratx=0.5,and explore the estimator’s graphical f it to the true underlying curve.We also investigate the goodness-of-f it to the normal distribution which is expected from our theoretical results in Section 2.At the same time,we check the influence of the dependence of the observations on the estimator.In order to obtain anα-mixing observed sequence{Xi,Zi,Ti,δi},we generate the observed data as follows.

(1)Drawing of the first observation(X1,Z1,T1,δ1)in the final sample.

Step 1Drawe1～N(0,1),and takeX1=0.5e1;

Step 2ComputeY1andW1,respectively,from the modelY1=sin(πX1)+φ1(1+0.3cos(πX1)),andW1=sin(πX1)+0.5φ2(1+0.3cos(πX1))+φ3(1+0.3cos(πX1))where bothandareN(0,1)random variables,andX1are mutually independent,andφi(i=1,2,3)are chosen(see below)to control the percentage of censoring.TakeZ1=min(Y1,W1),δ1=I(Y1≤W1);

Step 3Draw independentlyT1～N(μ,1),whereμis adapted in order to get different values ofθ.IfZ1

(2)Drawing of the second observation(X2,Z2,T2,δ2)in the final sample.

Step 4DrawX2according to the AR(1)modelX2=ρX1+0.5e2,wheree2～N(0,1)is independent ofX1,and|ρ|<1 is some constant,which is chosen to control the dependence of the observations;

Step 5ComputeY2andW2,respectively,from the modelY2=sin(πX2)+φ1(1+0.3cos(πX2))andW2=sin(πX2)+0.5φ2(1+0.3cos(πX2))+φ3(1+0.3cos(πX2))where bothandareN(0,1)random variables,andandX2are mutually independent.TakeZ2=min(Y2,W2),andδ2=I(Y2≤W2);

Step 6Draw independentlyIfZ2

By replicating the process(2)above,we generate the observed data(Xi,Zi,Ti,δi),i=1,···,n.The generating process shows thatXi=ρXi?1+0.5ei,Yi=sin(πXi)+φ1(1+0.3cos(πXi))i,Wi=sin(πXi)+0.5φ2(1+0.3cos(πXi))+φ3(1+0.3cos(πXi))Zi=min(Yi,Wi),andδi=I(Yi≤Wi),whereei～N(0,1),～N(0,1),andTi～N(μ,1);everything is distributed conditionally onZi≥Ti.Note that theα-mixing property of the observableXiis immediately transferred to the(Xi,Zi,Ti,δi).Also note that(sin(πx),(1+0.3cos(πx))2),which shows that the conditional quantile functionξ0.5(x)=sin(πx).For the proposed estimators,we employ the kernelK(x)=1).

In addition,the parametersφi(i=1,2,3)allow for the control of the percentage of censoring(PC)which is given by

In the simulation below,we takeφ1=φ3=0.3.

3.1 Consistency

In this subsection,we draw random samples with sample sizesn=200,350 and 500,respectively,andρ=0.1,0.3 and 0.5,respectively,from the above model.In Table 1,we report the MSE of the estimatorwithp=0.5 atx=0.5,for several truncation rates,percentage of censoring,and choice of bandwidth based onM=1000 replications.

Table 1 Mean squared errors(MSEs)ofwith p=0.5 at x=0.5 along M=1000 Monte Carlo trials,for several truncation rates and percentage of censoring(PC).

Table 1 Mean squared errors(MSEs)ofwith p=0.5 at x=0.5 along M=1000 Monte Carlo trials,for several truncation rates and percentage of censoring(PC).

ρθPCn hn=0.3 hn=0.35 hn=0.4 0.1 30% 10% 200 0.7569×10?2 1.0060×10?2 1.3393×10?2 350 0.6330×10?2 0.8714×10?2 1.2988×10?2 500 0.5296×10?2 0.8240×10?2 1.2279×10?2 15% 200 0.7652×10?2 1.0360×10?2 1.3612×10?2 350 0.6173×10?2 0.8544×10?2 1.2915×10?2 500 0.5267×10?2 0.8128×10?2 1.1961×10?2 30% 200 0.8009×10?2 1.0503×10?2 1.3777×10?2 350 0.6018×10?2 0.8430×10?2 1.1996×10?2 500 0.5227×10?2 0.7935×10?2 1.1153×10?2 60% 10% 200 0.7349×10?2 0.9612×10?2 1.2418×10?2 350 0.5631×10?2 0.8356×10?2 1.1839×10?2 500 0.5183×10?2 0.8090×10?2 1.1085×10?2 15% 200 0.7542×10?2 0.9762×10?2 1.2800×10?2 350 0.5549×10?2 0.8325×10?2 1.1601×10?2 500 0.5096×10?2 0.7807×10?2 1.1025×10?2 30% 200 0.7628×10?2 1.0204×10?2 1.2688×10?2 350 0.5473×10?2 0.8273×10?2 1.1345×10?2 500 0.5028×10?2 0.7543×10?2 1.0329×10?2 90% 10% 200 0.7149×10?2 0.9355×10?2 1.1969×10?2 350 0.5471×10?2 0.7639×10?2 1.1414×10?2 500 0.4975×10?2 0.7385×10?2 1.0816×10?2 15% 200 0.7291×10?2 0.9478×10?2 1.2218×10?2 350 0.5509×10?2 0.7886×10?2 1.1160×10?2 500 0.5076×10?2 0.7317×10?2 1.0550×10?2 30% 200 0.7301×10?2 0.9537×10?2 1.2277×10?2 350 0.5722×10?2 0.7453×10?2 1.0503×10?2 500 0.4738×10?2 0.7159×10?2 0.9987×10?2 0.3 90% 30% 200 0.7368×10?2 0.9551×10?2 1.2312×10?2 350 0.5830×10?2 0.7719×10?2 1.0861×10?2 500 0.5047×10?2 0.7336×10?2 1.0107×10?2 0.5 90% 30% 200 0.7498×10?2 0.9797×10?2 1.2419×10?2 350 0.5983×10?2 0.8019×10?2 1.1099×10?2 500 0.5248×10?2 0.7496×10?2 1.0410×10?2

From Table 1,it is seen that(i)the MSE decreases as the sample sizenincreases;(ii)the accuracy of the estimator is greatly af f ected by the choice of the bandwidthhn,i.e.,higher values forhngive bad estimators;(iii)for the same sample size,the performance of the estimator is af f ected slightly by the percentage of truncated data 1?θand the percentage of censoring PC;(iv)the values of the MSE become bigger as the dependence of the observations increases,i.e.,the value ofρincreases.

In Figures 1–3,we plot the Boxplots of the MSE for the estimatorwithp=0.5 andhn=0.3 atx=0.5,alongM=1000 Monte Carlo trials,forθ=90%,PC=30%,n=200,350 and 500;θ=90%,n=350,PC=10%,15%and 30%;PC=30%,n=350,θ=30%,60%and 90%,respectively.

Figure 1 shows that the quality of f it increases as the sample sizenincreases.

Figure 1 Boxplots ofwith p=0.5 and hn=0.3 at x=0.5 along M=1000 Monte Carlo trials,for θ=90%,PC=30%,n=200,350 and 500,respectively.

Figure 2 Boxplots ofwith p=0.5 and hn=0.3 at x=0.5 along M=1000 Monte Carlo trials,for θ=90%,n=350,PC=10%,15%and 30%,respectively.

From Figures 2–3,it can be seen that for the same sample size,the quality of the estimator does not seem to be af f ected by the percentage of truncated data 1?θand the percentage of censoring.

In Figure 4,we plot the averages of the curvesξp(x)=sin(πx)and its estimatorwithp=0.5 andhn=based on 100 replications forθ=90%,PC=10%,n=150,300 and 500,respectively.Figure 4 shows again that the quality of f it of the estimator increases as the sample sizenincreases.

3.2 Asymptotic normality

In this subsection,we examine how good is the asymptotic normality of the estimatorwithp=0.5 atx=0.5 by comparing the histograms and Normal-Probability-plots with the normal distribution.We drawMindependentn-samples.In Figures 5–6,we plot the histograms and Normal-Probability-plots forθ=90%,PC=10%andbased onM=1000 replications with sample sizesn=300 and 600,respectively.From Figures 5–6,it is seen that the sampling distribution of the estimator f its the normal distribution reasonably well;this f it being better when increasing the sample size.

Figure 3 Boxplots ofwith p=0.5 and hn=0.3 at x=0.5 along M=1000 Monte Carlo trials,for PC=30%,n=350,θ=30%,60%and 90%,respectively.

Figure 4 Function ξp(x)and its estimatorwith p=0.5 and hn=along M=100 Monte Carlo trials,for θ=90%,PC=10%,n=150,300 and 500,respectively.

Figure 5 Histogram and Normal-Probability-plot ofwith p=0.5 and hn=at x=0.5 along M=1000 Monte Carlo trials,for θ=90%,PC=10%,n=300.

To study the influence of the dependence of the observations,we consider different degrees of dependence;specifically we choose in Figure 7,ρ=0.1,0.3,0.5,respectively,and plot the Normal-Probability-plots ofwithp=0.5 andhn=atx=0.5 based onM=1000 replications withθ=90%,PC=30%,andn=400.Figure 7 shows that as the dependence of the observations increases,the quality of f it decreases.

Figure 6 Histogram and Normal-Probability-plot ofwith p=0.5 and hn=at x=0.5 along M=1000 Monte Carlo trials,for θ=90%,PC=10%,n=600.

Figure 7 Normal-Probability-plots ofwith p=0.5 and hn=at x=0.5 along M=1000 Monte Carlo trials,for θ =90%,PC=30%,n=400,ρ =0.1,0.3 and 0.5,respectively.

4 Proofs of the Main Results

Lemma 4.1Letx∈D and α(n)=O(n?λ)for some λ>6.Suppose that conditionsand(B1)–(B2)hold,and that τ1

(a)If(B3)(i)holds,then

(b)If(B3)(ii)holds,then

Proof of Lemma 4.1We prove only(a);the proof of(b)is similar.From Lemma 5.2,we have

Note that

Therefore

Note thatC(y|x)=θ?1(x)L(y|x)(1?G(y|x))(1?F(y|x))and=θ?1(x)(1?G(t|x))f(t|x)dt.ThenC(0,1)(y|x)andare bounded fory∈[τ1,τ2]from(A3’).Hence,using Lemmas 5.1–5.2,it follows that

Similarly,

and

Therefore,from(4.3)it follows that

Using Lemma 5.1,from(4.2)one can verify thata.s.

Therefore,

whereris betweensandt.Thus,the conclusion follows from(4.1)and(4.4)–(4.5).

Proof of Theorem 2.1We prove only(a);the proof of(b)is similar.Observe that

SinceF(·|x)is continuous,F(ξp(x)|x)=p.Then from the definition ofwe have

wherestands for the left-hand limit of

Since 0

Then,the first part of the theorem follows from Lemma 5.1 and(D1).Note that

where(x)is betweenξp(x)and(x).Then,by(4.8),we have

Thus,the second part of the theorem follows from Lemma 5.1 and(D2).

Proof of Theorem 2.2We prove only the conclusion in the case

fori=1,2.

Since=O(1),(b)in Theorem 2.1 ensures that

Therefore,using a Taylor expansion,it follows that

where(x)is between(x)andξp(x),and(ξp(x)|x)|=a.s.by Lemma 4.1.Hence fromf(ξp(x)|x)>0 andwe have

Note that(E1)implies thatis bounded.Then,according toa.s.from(b)in Theorem 2.1,it follows that

In addition,using Lemma 5.2 andF(ξp(x)|x)=p,we can write(4.9)as

a.s.from(D2’).

Proof of Theorem 2.3Note that→0 implies that→0.Then from Theorem 2.2 andF(ξp(x)|x)=p,we have

Therefore,from Lemma 5.3 it follows that

Proof of Theorem 2.4From Theorem 2.2 we write

LetThen,using Lemma 5.4 we have

From Lemma 5.1,it follows that

Lemma 5.1 and Theorem 2.2 ensure that

Let Λ(u)=E(ξ(Z,T,δ,ξp(x),x)|X=u,T≤Z).Then

Obviously,Λ(x)=0 and the function Λ has bounded the firstr0partial derivatives inU(x)from(A3’).Hence we have

Note thatThen from(4.10)–(4.13),it suffices to verify that

In fact,letw=and(Zi,Ti,δi,ξp(x),x).De fineymn(x),(x),(x)as follows:

wherekm=(m?1)(p+q)+1,lm=(m?1)(p+q)+p+1.Then

LetBy applying Lemma 5.4,it follows that

Then,to verify(4.14),we only need to prove that

and

(i)We verify(4.15).Note that

From(A1’)and(A3’),we get

Using(A1’)and(A4’),from the proof in(4.13)fori

On the other hand,from Lemma 5.5(takingp=q=20λ),it follows that

andE|ηi(x)|20λ≤=K20λ(s)m?(x?hns)ds =which yield|Cov(ηi(x),ηj(x))|≤C[α(j?i)]1?Letcn=for≤ρ<1.Then

Using Lemma 5.5 again,we have

From(4.17)–(4.20),we obtain(x))2=O(qp?1++γ3n)=O(τ1n)and

(ii)We prove(4.16).Letπmn(x),m=1,2,···,wbe independent random variables,where the distribution ofπmnis the same as that ofymn(x)form=1,2,···,w.PutUn=(x)and(x).Then

Note that

Then,in view ofm?(s)=θ?1θ(s)m(s),=θ?1(s)L(y|s)(1?G(y|s))f(y|s),andC(y|s)=θ?1(s)L(y|s)(1?G(y|s))(1?F(y|s)),from(A1’),(A3’)and(4.13),we have

Then,from(4.18)–(4.19)and(4.22),it follows that

which implies that→1 and

By the Berry-Esseen inequality(see[25,p.154,Theorem 5.7]),forl>2,there exists some constantC>0 such that

Takingl=2(1+β)andμ=δ?2β,we havel+μ=2+δ.Note thatβ≤implies thatλ≥Then,using Lemma 5.6(takep=landq=l+μ)andE|η1(x)|2+δ≤we have

which,together with(4.24),yields

Let?(t)andψ(t)be the characteristic functions ofandrespectively.By the Esseen inequality(see[25,p.146,Theorem 5.3]),for any Γ>0,

Using Lemma 5.7,we have

From(4.13)and|Cov(ηi(x),ηj(x))|≤Cminwe have

ThusH1n=From(4.25),we have

which yields thatH2n=Choose Γ=Then from(4.26),we have

Therefore,from(4.21),(4.23),(4.25)and(4.27),we have

5 Appendix

In this section,we list some preliminary lemmas which have been used in the proofs of the main results in Section 4.Let{χi,i≥1}be a stationaryα-mixing sequence of real random variables with mixing coefficients{α(k)}.

Lemma 5.1(see[20])Let α(n)=O(n?λ)for some λ>2,and let τ be a finite positiveconstant.SetΓ1n=max

(a)Suppose that(A1)–(A4)are satis fied.If(A5)(i)holds,thenO(Γ1n)a.s.If(A5)(ii)holds,then

(b)Letx∈D.Suppose thatare satisfied.If(i)holds,then(y|x)?F(y|x)|=Op(Γ1n).

If(ii)holds,thenOp(Γ1n),and(y|x)?C(y|x)|=Op(Γ1n).

Lemma 5.2(see[20])Set ξ(Z,T,δ,y,x)=Letx∈Dand α(n)=O(n?λ)for some λ>0.Suppose that conditionsand(B1)–(B2)hold,and that

(a)Let λ>6and τ1

where|Qn(y|x)|=O(Γ2n)a.s.when(B3)(i)holds;|Qn(y|x)|=Op(Γ2n)when(B3)(ii)holds.

(b)Let λ>4.If(B3)(i)holds,then

If(B3)(ii)holds,then

Lemma 5.3(see[20])Letx∈D and α(n)=O(n?λ)for some λ>6.Suppose that conditions(B1)–(B2),(B3)(ii)and(E2)–(E3)hold.If τ1

Lemma 5.4Let X,V and Y1,···,Ymbe random variables,and then for positive numbers a,w1,···,wm,we have|P(X≤uV)?Φ(u)|≤|P(X≤u)?Φ(u)|+P(|V?1|>a)+a and

ProofThe first inequality is a consequence of Michel and Pfanzagl[22]and the second one follows from Lemma 3.1 of Liang and Fan[17].

Lemma 5.5(see[11,Corollary A.2,p.278])Suppose that X and Y are random variables such that E|X|p<∞,E|Y|q<∞,where p,q>1,p?1+q?1<1.Then

Lemma 5.6(see[27,Theorem 4.1])Let20.Then there exists Q=Q(p,q,γ)<∞such that

Lemma 5.7(see[30])Let p and q be positive integers.1≤r≤w.If s>0,r>0withthen there exists a constant C>0such that

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