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Logo of nihpaAbout Author manuscriptsSubmit a manuscriptHHS Public Access; Author Manuscript; Accepted for publication in peer reviewed journal;
 
Aust N Z J Stat. Author manuscript; available in PMC 2010 August 16.
Published in final edited form as:
Aust N Z J Stat. 2010; 52(1): 75–91.
doi:  10.1111/j.1467-842X.2009.00567.x
PMCID: PMC2921901
NIHMSID: NIHMS180836

NONPARAMETRIC ESTIMATION OF CONDITIONAL CUMULATIVE HAZARDS FOR MISSING POPULATION MARKS

Summary

A new function for the competing risks model, the conditional cumulative hazard function, is introduced, from which the conditional distribution of failure times of individuals failing due to cause j can be studied. The standard Nelson–Aalen estimator is not appropriate in this setting, as population membership (mark) information may be missing for some individuals owing to random right-censoring. We propose the use of imputed population marks for the censored individuals through fractional risk sets. Some asymptotic properties, including uniform strong consistency, are established. We study the practical performance of this estimator through simulation studies and apply it to a real data set for illustration.

Keywords: competing risks, fractional risk set, Nelson–Aalen, right-censoring

1. Introduction

Consider a multistate model in which a healthy (state 0) subject may end up with one of J different types of failure. This model, called ‘competing risk’ in the statistical literature, is relevant in many disciplines, such as medicine, demography, actuarial science (as multiple decrement models), economics and manufacturing; see Crowder (2001) and Lindqvist (2006). For each unit, we observe both the time to failure T* and the type of failure X* [set membership] {1, …, J}. In this setup, the observed failure times are frequently right-censored, possibly owing to the termination of the study before all the subjects fail, or because subjects are lost to follow-up. Consequently, T* and X* are unknown for the right-censored individuals. To accommodate this situation, we include an additional random variable C, the latent time to censoring, which is observed when a subject is right-censored.

The study of competing risks considers the crude probability Fj(t) = Pr(T* ≤ t, X* = j), also known as the cumulative incidence function, and the cause-specific cumulative hazard function

Λj(t)=0tdFj(υ)1F(υ),
(1)

where F=j=1JFj is the cumulative distribution function (cdf) of T*. Neither function may be appropriate if one desires to understand the summary probability of the different causes of failure, given that failures have already occurred. For our study, we focus on the estimation of the jth conditional functions

Fj*(t)=Pr(T*t|X*=j)andΛj*(t)=0tdFj*(υ)1Fj*(υ).
(2)

This approach is suitable when the target is the complete distribution and not the subdistribution function of the lifetimes associated with each risk separately as in (1).

For a better explanation of this problem, we consider the following real-life example. A clinical trial of estrogen diethylstilbestrol (DES) was carried out in which patients with stage 3 or 4 prostrate cancer were assigned to four treatment groups (Cheng, Fine & Wei 1998; Escarela & Carrière 2003). Patients died of either ‘prostate cancer’ (j = 1) or ‘other causes’ (j = 2). Also considered were right-censored individuals, whose eventual cause of death was unknown, however the cause of death might be due to any one of the two causes. The conditional distribution function Fj* is more appropriate to answer the question: ‘Among all the patients who died of prostate cancer (in a situation where we would know the cause of failure of all the individuals), what is the chance of surviving to age 60?’ whereas, the crude probability Fj is the answer to the specific question: ‘What is the chance that a patient will die of prostate cancer before age 60?’. We revisit this data set in Section 5.

The rest of the paper is organized as follows. In Section 2, we introduce the Nelson–Aalen type estimator of the conditional cumulative hazard function based on fractional risk sets (FRSs), and in Section 3 we give some asymptotic properties, including a strong representation and consistency. In Section 4, we consider a small simulation study to assess the finite sample performance of the novel estimator, and apply it to a real data set in Section 5. We conclude with a discussion in Section 6, followed by an Appendix, which contains the proofs of the main results.

2. Estimation based on fractional risk sets

We consider the competing risk network as a multistate continuous-time stochastic process {Z(t), t [set membership] T} with a finite-state space S = {1, …, J, 0} having a tree topology and right-continuous sample paths Z(t+) = Z(t), where we assume that the states 1, …, J are absorbing whereas state 0 is transient (the root node). Here T = [0, τ], where τ is a large, possibly observed, time point (≤ ∞). Typically, for applications, τ will be taken to be the largest time at which some event (failure) took place. Let Ti* be the time at which the ith individual leaves stage 0 for a failure (stage j, say), with cumulative distribution function (cdf) F, and let Xi* denote the stage occupied by the ith individual at time Ti* (i.e., its failure type). In the presence of right-censored observations, we record the censoring time C with cdf G. Hence, although the variables of interest are T* and X*, one cannot observe (Ti*,Xi*), but (Ti, δi, Xi), with Ti=min(Ti*,Ci), the right-censored failure time with cdf H, the censoring indicator δi=1(Ti*Ci), and the observable cause of failure Xi=Xi*δi. If an individual is not censored, then Xi*=Xi; otherwise, Xi = 0 and X* is unknown. It is further assumed that the censoring variable C is independent of (T*, X*), and that all the random variables are independent and identically distributed (i.i.d.) across the n individuals.

There are no data-driven estimators of Λj* and Fj* in the literature. One may be tempted to estimate these functions using classical ideas, as in Andersen et al. (1993):

Λ^j*(t)=0t1(Yj(υ)>0)Yj(υ)dNj(υ)=Tit1(Xi=j)Yj(Ti)andF^j*(t)=1Tit(11(Xi=j)Yj(Ti)),

where Nj(t)=i=1n1(Tit,Xi=j) is the number of observed failures of type j in the time interval [0, t], and

Yj(t)=i=1n1(Tit,Xi*=j)
(3)

is the number of individuals at risk of failing due to cause j or of being censored. The functions Λ^j* and F^j* are not estimators themselves, as the size of the subpopulation at-risk set Yj is not computable with the observations (Ti, δi, Xi). In the absence of such an identifier, we can still assign a probability of each individual being in one of the J subpopulations. We estimate Yj by Yjf, the ‘fractional risk set’ corresponding to the jth cause of failure (Satten & Datta 1999; Datta & Satten 2000), defined as

Yjf(t)=i=1nϕ^ij1(Tit).
(4)

Here, [phi with hat]ij is the estimated probability that the ith individual belongs to the jth subpopulation according to its failure type:

ϕ^ij={1,ifXi=j0,ifXi>0,Xij,P^j(Ti,),ifXi=0

where

P^j(s,t)=(s,t]((s,u)(1dN(υ)Y(υ)))dNj(u)Y(u),
(5)

with N(t)=j=1JNj(t) and Y(t)=i=1n1(Tit), is the Aalen–Johansen estimator (Andersen et al. 1993, pp. 288–289), and the nonparametric maximum likelihood estimator (NPMLE) of the transition probability is

Pj(s,t)=P(T*t,X*=j|T*>s)=P(Failure type j by timet|  alive at time s).
(6)

Then, Yjf(t) gives the estimated fractional mass in the jth failure group remaining at risk of failure at time t, counting ([phi with hat]ij = 1) the observations that failed due to cause j, discarding ([phi with hat]ij = 0) the observations that failed due to a cause other than j, and estimating the probability of being in the jth failure-type group ([phi with hat]ij = Pj(Ti, ∞)) for the censored observations. Thus, the proposed estimators of Λj* and Fj* are

Λ^j*f(t)=Tit1(Xi=j)Yjf(Ti)andF^j*(t)=1Tit(11(Xi=j)Yjf(Ti)),

with Yjf given in (4). In fact, the proportion πj = Pr(X* = j) of the sample that fails due to cause j can be estimated as

P^j(0,)=1ni=1nϕ^ij,j=1,,J.

The ‘fractional risk set’ concept has recently been used in the literature with multistate models; see Satten & Datta (1999), Datta & Satten (2000) and Bandyopadhyay & Datta (2008).

3. Asymptotic properties

As a first step, we provide some results for Pj and Yjf, which are of independent interest in themselves. They will be used to derive the main properties of Λ^j*f. The analogous results for F^j*f can be derived considering the one-to-one mapping relation

1F(t)=exp(Λc(t))ut(1ΔΛ(u)),

where Λc is the continuous part of Λ, and ΔΛ(u) = Λ(u) – Λ(u). We first obtain, in Theorem 1, an i.i.d. representation of Pj. This is a novel result that can be used to derive many other asymptotic properties of Pj, for example the strong consistency result in Corollary 1, which is analogous to the consistency in Aalen (1978, theorem 1) for partial transition probabilities and to that in Fleming (1978, theorem 5.1) for nonhomogeneous Markov processes. The same properties will be obtained, in Theorem 2 and Corollary 2 respectively, for the fractional risk-set estimator Yjf.

From these properties for Pj and Yjf, and the relationship

Λ^j*f(t)=Λ^j*(t)+0t(1Yjf(υ)1Yj(υ))dNj(υ),
(7)

we obtain the strong i.i.d. representation of Λ^j*f in Theorem 3, and the uniform strong consistency in Corollary 3.

Before we state the main results, we introduce some notation. Fix bH < sup{t : H(t) < 1}, with H the cdf of the observable lifetime T, and consider the subdistribution functions (c, censored; nc, not-censored)

Hnc(t)=Pr(Tt,δ>0),
(8)

Hc(t)=Pr(Tt,δ=0)andH¯c(t)=Pr(Tt,δ=0),
(9)

Hjnc(t)=Pr(T*t,δ>0,X*=j)=Pr(Tt,X=j),
(10)

Hj(t)=Pr(Tt,X*=j)  and  H¯j(t)=Pr(Tt,X*=j).
(11)

Consider now the empirical estimators of (10) and (11) given by

Hnjnc(t)=1ni=1n1(Tit,Xi=j)=1nNj(t),Hnj(t)=1ni=1n1(Ti<t,Xi*=j)=1n(njYj(t)),

where nj=i=1n1(Xi*=j). The importance of these functions is clear, as the Nelson–Aalen type estimator Λ^j*(t) of the conditional cumulative hazard function can also be written as follows:

Λ^j*(t)=0tdHnjnc(υ)nj/nHnj(υ).

The following theorem gives an i.i.d. representation of the estimator of the transition probabilities Pj.

Theorem 1. If the distribution functions F and G are continuous, then

P^j(s,t)Pj(s,t)=1ni=1nζj(Ti,Xi*,δi,s,t)+rn(s,t),

where

ζj(T,X*,δ,s,t)=11F(s)×(1F(T)1H(T)1(sTt,δ>0,X*=j)st1F(υ)(1H(υ))21(Tυ,X*=j)dHnc(υ)11H(T)st(1F(υ))1(sTυ,δ>0)dΛj(υ)+st(1F(υ))0υ1(Tu)(1H(u))2dHnc(u)dΛj(υ))Pj(s,t)0s1(Tυ)(1H(υ))2dHnc(υ),
(12)

with Hnc given in (8) and sup0≤s≤t≤bH |rn(s, t)| = O(n−1 ln n) almost surely.

Remark 1. This result generalizes the i.i.d. representation for the Aalen–Johansen estimator evaluated at (t, ∞), which is important in the study of many statistical properties in competing risks:

P^j(t,)Pj(t,)=1ni=1n11F(t)t1F(υ)1H(υ)×(dMji(υ)Pj(υ,)dMi(υ))+rn(t),

where

Mji(t)=1(Tit,δi>0,Xi*=j)0t1(Tis,Xi*=j)dΛ(s),Mi(t)=j=1JMji(t)

and sup0≤t≤bH |rn(t)| = O (n−1 ln n) almost surely.

Corollary 1. The estimator of the transition probabilities satisfies

sup0stbHn1/2(lnn)1/2|P^j(s,t)Pj(s,t)|0.

Theorem 2. If the distribution functions F and G are continuous, then

1n(Yj(t)Yjf(t))=1ni=1nρj(Ti,Xi*,δi,t)+sn(t),

where

ρj(T,X*,δ,t)=1(Tt,δ=0)(1(X*=j)Pj(T,))tζj(T,X*,δ,υ,)dH¯c(υ),
(13)

with Hc given in (9), ζj in (12) and sup0≤t≤bH |sn(t)| = O(n−1 ln n) almost surely.

Corollary 2. The fractional risk-set estimator Yjf satisfies

sup0tbHn1/2(lnn)1/2|Yj(t)Yjf(t)|0.

The following theorem gives a representation of the FRS estimator of Λj* as a sum of i.i.d. variables plus a remainder term. It is based on the strong representations for the transition probabilities in Theorem 1 and for the FRS in Theorem 2.

Theorem 3. If the distribution functions F and G are continuous, then

Λ^j*f(t)Λj*(t)=1ni=1nξj(Ti,Xi*,δi,t)+Rn(t),

where

ξj(T,X*,δ,t)=1(Tt,X*=j)H¯j(T)0t1(Tυ,X*=j)H¯j2(υ)dHjnc(υ)+0tρj(T,X*,δ,υ)H¯j2(υ)dHjnc(υ)

with Hjnc given in (10), Hj in (11), ρj in (13), and

sup0tbH|Rj(t)|=O(n1(lnn)3)almost surely.

Corollary 3. The fractional risk-set estimator of Λj* satisfies

sup0tbH(n(lnn)3)1/2|Λ^j*f(t)Λj*(t)|0almost surely.

Remark 2. The nature of the dependence between T* (time to failure) and X* (cause of failure) is very useful. Under independence, both variables can be studied separately, which greatly simplifies the analysis of competing risks. There are few tests in the literature for the independence between T* and X*. Without censoring, Dewan, Deshpande & Kulathinal (2004) proposed several tests based on the conditional probabilities [var phi]j(t) = Pr(X* = j | T* > t) = Pj(t, ∞), as T* and X* are independent if and only if [var phi]j is a constant.

With censored observations, Dykstra, Kochar & Robertson (1998) and Kochar & Proschan (1991) provided some restricted tests. More recently, tests of independence using FRSs were studied by Bandyopadhyay & Datta (2008). In this context, the conditional functions Fj* and Λj* could be used to test independence without any restriction as, in that case, Fj*=F and Λj*=Λ.

4. Simulation study

We carried out a small simulation study to assess the practical performance of the FRS estimator of Λj*. Note that, when T* and X* are independent, we have Λj*=Λ for j = 1, …, J. So, we also considered the Nelson–Aalen estimator of Λ,

Λ^nNA(t)=Tit1(δi>0)Y(Ti)  with  Y(Ti)=k=1n1(TkTi),

to estimate Λj* under the independence of T* and X*. However, the independence assumption is not always appropriate, and Λ^j*f will be the only available estimator of Λj*.

We simulated two models with J = 2 competing risks. For Model 1 (El-Nouty & Lancar 2004), we assume the independence of T* and X*, whereas in Model 2 the variables T* and X* are dependent. In both models, T* is the lifetime (in days) and takes values in the interval [0, τ]. The parameter τ was chosen equal to 1461 to represent 4 years. In this case, the percentage of censoring is about 8.33% in Model 1 and 33% in Model 2.

Model 1. The variables T* and X* are independent, with

F(t)=1(1t/τ)3/4 exp(t2/(2τ2))  and  G(t)=1(1t/τ)1/4 exp(t2/(2τ2))F1(t)=F2(t)=0.5F(t)Λj*(t)=t2/(2τ2)3/4 ln (1t/τ)forj=1,2.

Model 2. The variables T* and X* are dependent, with

F(t)=t/τ and G(t)=1(1t/τ)2F1(t)=t/(2τ)(1(1t/τ)2) and  F2(t)=t/(2τ)(1+(1t/τ)2)Λ1*(t)=ln(1t/τ(1(1t/τ)2))  and  Λ2*(t)=ln(1t/τ(1+(1t/τ)2)).

For Model 1, [var phi]j(t) = P(X* = j) = 0.5 for j = 1, 2, and therefore the variables T* and X* are independent. However, for Model 2, we have

φ1(t)=0.5(1+tτt2τ2)  and φ2(t)=0.5(1tτ+t2τ2),

and, as [var phi]j is not constant, T* and X* are dependent.

Figure 1 shows the FRS estimator Λ^j*f for j = 1, 2 and the NA estimator Λ^nNA. Under the independence of T* and X* (Model 1), we have Λ1*=Λ2*=Λ, and the three estimators are very similar. However, when T* and X* are dependent (Model 2), Λ^j*f and Λ^nNA differ from each other. Hence, the plot of the FRS and NA estimators can be used as a preliminary test for the independence of T* and X*.

Figure 1
Fractional risks set (FRS) estimators of Λj* (lines with empty boxes, j = 1, and with solid boxes, j = 2) and Nelson–Aalen (NA) estimator of Λ (plain line) for Model 1 (under independence of T* and X*) and Model 2 (when T* and ...

Under independence, the best estimator for Λj* is Λ^nNA rather than Λ^j*f, j = 1, 2. In order to show this, we computed the mean squared error (MSE) of the FRS and NA estimators as (see Figure 2)

MSE(Λ^j*f(t))=E((Λ^j*f(t)Λj*(t))2)  and  MSE(Λ^nNA(t))=E((Λ^nNA(t)Λj*(t))2).
Figure 2
Mean squared error (MSE) of the fractional risk set (FRS) estimators of Λj* (lines with empty boxes, j = 1, and with solid boxes, j = 2) and Nelson–Aalen (NA) estimator of Λ (plain line, j = 1, and dashed line, j = 2) when T* and ...

In Model 1 (T* and X* are independent), the NA estimator is preferable for estimating Λj* as it has a smaller MSE. However, under dependence (Model 2), MSE(Λ^nNA) is larger than MSE(Λ^j*f). Hence, we infer that the NA estimator is not a serious candidate for estimating Λj* unless there is clear evidence for the independence of T* and X*, and the FRS estimator should be used in any case.

The FRS-based estimators can also be used to check the independence of T* and X* using a Kolmogorov–Smirnov (KS)-type test. Under independence, we have Fj*=F, j = 1, …, J. Thus, considering the FRS estimator of Fj* and the Kaplan–Meier (KM) estimator of F, we construct the following KS-type test:

TKS=suptmaxj=1,,J|F^j*f(t)FnKM(t)|.

Figure 3 provides the histogram of the values of the KS test computed using m = 1000 Monte Carlo samples of size n = 100. As expected, TKS takes much lower values in Model 1, where T* and X* are independent, than in Model 2. This shows that the KS test is promising for testing independence. However, the theoretical large-sample properties of this test, together with size and power evaluations, are outside the scope of this paper and will be addressed in the future.

Figure 3
Values of the Kolmogorov–Smirnov test TKS for Model 1 (under H0, T* and X* are independent) and Model 2 (under H1).

5. Example: prostate cancer data

In this section, we illustrate the FRS estimators using the prostate cancer data described in Section 1. We consider the same data set as studied by Byar & Green (1980) and published in Andrews & Herzberg (1985). In those papers, the randomized trial was aimed at comparing different levels of DES, a drug used to treat prostate cancer, with respect to patient survival. A total of 506 patients were assigned to four treatment groups. Because of the potentially fatal cardiovascular adverse effect from DES (Escarela & Carrière 2003), the assessment of risk–benefit analysis of DES (Cheng et al. 1998) must take into account not only the death time from prostate cancer (j = 1), but also other competing causes of death (j = 2), including death from cardiovascular-related causes. Although we do not consider the set of covariates, 23 patients with incomplete covariate information were removed from our analysis. Out of the 483 patients, there were 125 (about 26%) deaths from prostate cancer, 219 (about 45%) deaths from ‘other causes’, along with 139 (about 29%) right-censored observations whose subpopulation membership of the eventual cause of failure was unknown, however it was assumed to be one of the two causes for our problem. We are interested in the distribution of the time to death from prostate cancer (j = 1) and other different causes (j = 2), and also in verifying the dependence/independence between the lifetime and the cause of failure.

First, we estimate the distribution functions Fj* using the FRS estimator. Figure 4 shows that, among all the patients who died because of prostate cancer (including censored individuals who may die of prostate cancer), the chance of surviving to month 60 after administration of DES is 0.87. Note that the FRS estimates of Fj* for j = 1 and j = 2 are very similar, which suggests that the lifetime and cause of failure are independent.

Figure 4
Fractional risk-set (FRS) estimators of the distribution functions for the two competing risks, prostate cancer (j = 1) and other causes (j = 2), in the Prostate cancer data set.

We also computed the FRS estimator of Λj* for j = 1 and 2 and the NA estimator Λ^nNA. Figure 5 shows that the FRS estimators are very close to each other, and quite different from Λ^nNA. This agrees with the conclusion of the log-rank test defined in Bandyopadhyay & Datta (2008) that tests the null hypothesis H0:F1*(t)==FJ*(t), which is equivalent to testing the independence of T* and X*. The log-rank test statistic for this data set is 5.989, with a bootstrap estimated standard error of 14.275, leading to a p-value of 0.176, which is not significant at the 5% level. We can conclude that, among the patients who died of (a) prostate cancer and (b) other causes, the application of the drug DES did not provide any differential effect on the survival behaviour of the two competing groups.

Figure 5
Fractional risk-set (FRS) and Nelson–Aalen (NA) estimators of the cumulative hazard functions for the two competing risks, prostate cancer (j = 1) and other causes (j = 2), in the Prostate cancer data set.

6. Discussion

In this article, two new functions, the jth conditional distribution function Fj* and the corresponding cumulative hazard function Λj*, were introduced for the competing risks model. This helps us to describe the distribution of the lifetimes of those individuals failing from a specific cause j when there is a set of J interacting causes.

A new Nelson–Aalen-type estimator of Λj* was proposed for when the population marks of some individuals are unknown owing to right-censoring. The key is to split the total risk set at every time point t into J possible at-risk subsets, also called the FRS. The FRS methodology provides a way to include the censored observations in the estimation with a nice probability interpretation. Some properties of the FRS estimator Λ^j*f have been derived, such as an asymptotic i.i.d. representation and a strong consistency result, together with the same results for the transition probability estimator Pj and the FRS estimator Yjf. Our simulation studies show that, in addition to providing information on lifetimes of individuals failing due to a specific cause, FRS estimators of Fj* and Λj* can be used to test the independence of T* and X*.

Although we have restricted our attention to the estimation of the conditional cumulative hazards under a competing risk framework, this can be extended much further to more complicated multistate networks, for example the three-stage irreversible illness–death model (Andersen et al. 1993, pp. 28–34). The conditional cumulative hazards among different states in the multistate model can be compared using FRSs. This will be pursued elsewhere.

Acknowledgments

Part of this work was done while the first author was a doctoral student at the University of Georgia, USA. He acknowledges research support from the University of Georgia through a Graduate School Dissertation Completion Fellowship. This research was partially supported by grants P20 RR017696-06 and 5U10 DA013727-09 from the United States National Institutes of Health. The second author would like to acknowledge economic support through grant MTM2008-00166 from the Spanish Ministerio de Ciencia e Innovación. The authors thank Professor Somnath Datta for his valuable comments and for drawing attention to the fractional risk-set methodology. We are also grateful to the Associate Editor and a referee for their critical comments, which led to a substantially better presentation of the manuscript.

Appendix

Proof of Theorem 1. The transition probabilities in (6) can also be written as follows:

Pj(s,t)=11F(s)st(1F(υ))dΛj(υ),

with F the cdf of T*, and Λj the cause-specific cumulative hazard function given in (1). For the Aalen–Johansen estimator of Pj(s, t) in (5), we have

P^j(s,t)=11FnKM(s)st(1FnKM(υ))dΛ^jNA(υ),

where FnKM is the product-limit Kaplan–Meier (KM) estimator, and Λ^jNA the Nelson–Aalen (NA) estimator of Λj. Then the i.i.d. representation of Pj(s, t) – Pj(s, t) comes from that of FnKM (Zhou & Yip 1999) and Λ^jNA (Lo, Mack & Wang 1989).

Proof of Corollary 1. Applying the strong uniform consistency results for the KM and NA estimators (Zhou & Yip 1999, and Lo et al. 1989), the proof is straightforward.

Proof of Theorem 2. Recall the definition of the risk set Yj in (3) and the FRS Yjf in (4). Then

1n(Yj(t)Yjf(t))=1ni=1n(1(Xi*=j)Pj(Ti,))1(Tit,δi=0)+1ni=1n(Pj(Ti,)P^j(Ti,))1(Tit,δi=0).
(14)

The first term is a normalized sum of zero-mean summands (see Datta & Satten 2000, lemma 3.1). The second term in (14) can be written as follows:

tbH(P^j(υ,)Pj(υ,))dH¯nct(υ)  =1ni=1ntbHζj(Ti,Xi,δi,υ,)dH¯c(υ)+sn(t),

where H¯nc is the empirical estimator of Hc in (9), ζj is given in (13), and sup0≤t≤bH |sn(t)| = O(n−1 ln n).

Proof of Corollary 2. The triangle inequality allows us to bound the first term in the decomposition (14) by the sum

|1ni=1n1(Tit,δi=0)1(Xi*=j)Pr(Tt,δ=0,X*=j)|+|1ni=1n1(Tit,δi=0)Pr(Tt,δ=0)|Pr(X*=j|δ=0).

These summands are the absolute error of estimation of some empirical distribution functions. An immediate consequence of the Dvoretzky–Kiefer–Woldfowitz (DKW) bound for empirical measures is that the supremum of the first term in (14) is O(n−1/2(ln n)1/2) almost surely. The second term in (14) can be written as follows:

tbH(P^j(υ,)Pj(υ,))1(υt)dHnc(υ),

with Hnc the empirical estimator of Hc in (9). Then, applying Corollary 1, the supremum of the second term in (14) is o(n−1/2 ln n) almost surely.

The outline of the proof of Theorem 3 is similar to that of Major & Rejtö (1988, theorem 1). We start with a few preliminary results that will be useful for some methods in the main body of the proof of Theorem 3 and Corollary 3. Consider the function Hj in (11) and its empirical estimator

H¯nj(t)=1ni=1n1(Tit,Xi*=j).

The following lemmas give some consistency results for Hnj.

Lemma 1. sup0tbH|H¯j(t)H¯nj(t)|=O(n1/2(lnn)1/2) a.s.

Proof. It is an immediate consequence of the DKW bound for empirical measures.

Lemma 2. supk:TkbHH¯j(Tk)/H¯nj(Tk)=O(lnn)  a.s.

Proof. The proof follows the same steps, in the two-dimensional case, as in Shorack & Wellner (1986, p. 415).

Proof of Theorem 3. From the definition of Λj* in (2) and the representation (7) of the FRS estimator, we have

Λ^j*f(t)Λj*(t)=0td(Hnjnc(υ)Hjnc(υ))H¯j(υ)0tH¯nj(υ)H¯j(υ)H¯j2(υ)dHjnc(υ)+Rj1(t)+Rj2(t)+Rj3(t),
(15)

where Hnjnc and Hnj are the empirical estimators of Hjnc and H j given in (10), and

Rj1(t)=0tH¯j(υ)H¯nj(υ)H¯j2(υ)d(Hnjnc(υ)Hjnc(υ)),Rj2(t)=0t(H¯j(υ)H¯nj(υ))2H¯nj(υ)H¯j2(υ)dHnjnc(υ),Rj3(t)=0t1(Yj(υ)>0)(1Yjf(υ)1Yj(υ))dNj(υ).

The term Rj1 can be decomposed into four terms:

Rj1(t)=0tdHnjnc(υ)H¯j(υ)0tH¯nj(υ)H¯j2(υ)dHnjnc(υ)0tdHjnc(υ)H¯j(υ)+0tH¯nj(υ)H¯j2(υ)dHjnc(υ).
(16)

The second integral in (16) is a V-statistic of order two. We work with it as follows:

0tH¯nj(υ)H¯j2(υ)dHnjnc(υ)=n1n0tdHnjnc(υ)H¯j(υ)+n1n0tH¯nj(υ)H¯j2(υ)dHjnc(υ)n1n0tdHjnc(υ)H¯j(υ)+Qn(t),
(17)

where we split up the integral into two terms, its diagonal and off-diagonal parts, and obtain the Hájek projection of the U-statistic. The term Qn(t) is a degenerate U-statistic of order two, and then (see Serfling 1980, section 5.3.3), for each δ > 3/2,

sup0tbH|Qn(t)|=o(n1lnn) almost surely.

Application of the strong law of large numbers (SLLN) to each of the remaining processes in (17) allows us to replace (n – 1)/n by 1, so that from (16) we have

sup0tbH|Rj1(t)|=o(n1lnn)  with probability  1.
(18)

An immediate consequence of Lemmas 1 and 2 and the SLLN is that

sup0tbH|Rj2(t)|=O(n1(lnn)2)  with probability  1.
(19)

For Rj3, consider the decomposition

Rj3(t)=n10tYj(υ)Yjf(υ)H¯j2(υ)dHjnc(υ)+n10t(Yj(υ)Yjf(υ))(1n2Yjf(υ)Yj(υ)1H¯j2(υ))dHjnc(υ)+n0tYj(υ)Yjf(υ)Yjf(υ)Yj(υ)d(HnjncHjnc)(υ).
(20)

The last term in (20) is, by Corollary 2, Lemma 2 and the DKW bound for empirical measures, O(n−1(ln n)3) almost surely. On applying Corollary 2, Lemma 1 and the DKW bound to the second integral in (20), it is O(n−1 ln n) almost surely. For the first integral in (20), the dominant term comes from the result in Theorem 2,

1ni=1n0tρj(Ti,Xi*,δi,υ)H¯j2(υ)dHjnc(υ)+0tsn(υ)H¯j2(υ)dHjnc(υ),

with sup0≤t≤bH |sn(t)| = O(n−1 ln n) almost surely. Therefore,

Rj3(t)=1ni=1n0tρj(Ti,Xi*,δi,υ)H¯j2(υ)dHjnc(υ)+Rj4(t)

with

sup0tbH|Rj4(t)|=O(n1lnn)  almost surely
(21)

The proof is completed using the decomposition (15) and the rates (18), (19) and (21).

Proof of Corollary 3. Consider the decomposition

Λ^j*f(t)Λj*(t)=0t(1H¯nj(υ)1H¯j(υ))dHnjnc(υ)+0td(Hnjnc(υ)Hjnc(υ))H¯j(υ)+0t(1Yjf(υ)1Yj(υ))dNj(υ).
(22)

It follows, from Lemmas 1 and 2 and the SLLN, that the absolute value of the first term in (22) is O(n−1/2(ln n)3/2) almost surely.

For the second term in (22), we apply integration by parts and the DKW bound for empirical measures. Then, this term is O(n−1/2(ln n)1/2) almost surely.

The third term in (22) can be written as follows:

n0tYj(υ)Yjf(υ)n2Yj2(υ)dHnjnc(υ)n0tYj(υ)Yjf(υ)n2Yj2(υ)Yj(υ)(Yjf(υ)Yj(υ)1)dHnjnc(υ).
(23)

The second term in (23) is negligible with respect to the first one. Therefore, from Corollary 2, Lemma 2 and the SLLN, the second term in (23) is O(n−1/2 ln n) almost surely.

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