Strong representation results of the Kaplan-Meier estimator for censored negatively associated data
© Wu and Chen; licensee Springer 2013
Received: 4 February 2013
Accepted: 10 July 2013
Published: 25 July 2013
In this paper, we discuss the strong convergence rates and strong representation of the Kaplan-Meier estimator and the hazard estimator based on censored data when the survival and the censoring times form negatively associated (NA) sequences. Under certain regularity conditions, strong convergence rates are established for the Kaplan-Meier estimator and the hazard estimator, and the Kaplan-Meier estimator and the hazard estimator can be expressed as the mean of random variables, with the remainder of order a.s.
KeywordsNA sequence random censorship model Kaplan-Meier estimator strong representation strong convergence rate
1 Introduction and main results
where denote the order statistics of , and is the concomitant of .
There is extensive literature on the Kaplan-Meier and the hazard estimator and for censored independent observations. We refer to papers by Breslow and Crowley , Foldes and Rejto  and Gu and Lai . Martingale methods for analyzing properties of are described in the monograph by Gill . However, the censored dependent data appear in a number of applications. For example, repeated measurements in survival analysis follow this pattern, see Kang and Koehler  or Wei et al. . In the context of censored time series analysis, Shumway et al.  considered (hourly or daily) measurements of the concentration of a given substance subject to some detection limits, thus being potentially censored from the right. Ying and Wei , Lecoutre and Ould-Saïd , Cai  and Liang and Uña-Álvarez  studied the convergence of for the stationary α-mixing data.
The main purpose of this paper is to study the strong convergence rates and strong representation of the Kaplan-Meier estimator and the hazard estimator based on censored data when the survival and the censoring times form the NA (see the following definition) sequences. Under certain regularity conditions, we find strong convergence rates of the Kaplan-Meier and hazard estimator, and the expression of the Kaplan-Meier estimator and the hazard estimator as the mean of random variables, with the remainder of order a.s.
where and are increasing for every variable (or decreasing for every variable) so that this covariance exists. A sequence of random variables is said to be NA if every finite subfamily is NA.
Obviously, if is a sequence of NA random variables, and is a sequence of nondecreasing (or non-increasing) functions, then is also a sequence of NA random variables.
This definition was introduced by Joag-Dev and Proschan . A statistical test depends greatly on sampling. The random sampling without replacement from a finite population is NA, but is not independent. NA sampling has wide applications such as in multivariate statistical analysis and reliability theory. Because of the wide applications of NA sampling, the limit behaviors of NA random variables have received more and more attention recently. One can refer to Joag-Dev and Proschan  for fundamental properties, Matula  for the three series theorem, and Wu and Jiang [15, 16] for the strong convergence.
We give two lemmas, which are helpful in proving our theorems.
Lemma 1.1 (Yang , Lemma 1)
Proof Similar to the proof of Lemma 4 in Yang , we can prove Lemma 1.2. □
here and in the sequel, .
where a.s. , .
follow from Lemma 1.2 and the fact that both and are empirical distribution functions of L and .
Thus, (1.5) holds.
Thence, the combination (1.5), (1.6) holds. This completes the proof of Theorem 1.3. □
For , , , let , . Then , and are NA sequences with , , , , .
Using the bound and the Borel-Cantelli lemma, we deduce that a.s. The estimation of is similar noting that for all x and y. Therefore, by (2.6)-(2.9), (1.8) holds. (1.9) follows from (2.5) and (1.8). □
Qunying Wu, Professor, Doctor, working in the field of probability and statistics.
Supported by the National Natural Science Foundation of China (11061012), project supported by Program to Sponsor Teams for Innovation in the Construction of Talent Highlands in Guangxi Institutions of Higher Learning ( 47), and the Support Program of the Guangxi China Science Foundation (2012GXNSFAA053010, 2013GXNSFDA019001).
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