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

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.