The Annals of Statistics

Empirical Processes Associated with $V$-Statistics and a Class of Estimators Under Random Censoring

Michael G. Akritas

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Abstract

A class of empirical processes associated with $V$-statistics ($V$-empirical process) under random censoring, and a class of nonparametric estimators based on the corresponding quantile process are defined. The $V$-empirical process is the censored data analogue of the $U$-empirical process considered by Silverman (1976, 1983). The class of estimators is the analogue of the class of generalized $L$-statistics introduced by Serfling (1984) and it includes the results of Sander (1975). The weak convergence of the $V$-empirical process and the corresponding quantile process is obtained and, through that, the asymptotic behavior of the estimators is studied. Linear bounds for the Kaplan-Meier estimator near the origin are established. A number of examples are given, including the generalization of the Hodges-Lehmann estimator for estimating the treatment effect in the two-sample problem under random censoring. A measure of spread, a procedure for estimation in the two-way ANOVA model, and a modified version of the two-sample Hodges-Lehmann estimator, all of which are new even in the uncensored case, are proposed.

Article information

Source
Ann. Statist., Volume 14, Number 2 (1986), 619-637.

Dates
First available in Project Euclid: 12 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aos/1176349942

Digital Object Identifier
doi:10.1214/aos/1176349942

Mathematical Reviews number (MathSciNet)
MR840518

Zentralblatt MATH identifier
0603.62046

JSTOR
links.jstor.org

Subjects
Primary: 62G05: Estimation
Secondary: 62G30: Order statistics; empirical distribution functions 62E20: Asymptotic distribution theory

Keywords
$V$-statistics empirical processes random censoring $L$-statistics Kaplan-Meier estimator Hodges-Lehmann estimator

Citation

Akritas, Michael G. Empirical Processes Associated with $V$-Statistics and a Class of Estimators Under Random Censoring. Ann. Statist. 14 (1986), no. 2, 619--637. doi:10.1214/aos/1176349942. https://projecteuclid.org/euclid.aos/1176349942


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Corrections

  • See Correction: Michael G. Akritas. Correction: Empirical Processes Associated with $V$-Statistics and A Class of Estimators Under Random Censoring. Ann. Statist., Volume 17, Number 3 (1989), 1417--1417.