Some asymptotic results for kernel density estimation under random censorship

Abstract

Random censored data consist of i.i.d. pairs of observations (X_i,δ_i), i=1,...,n. If δ_i=0, X_i denotes a censored observation, and if δ_i=1, X_i denotes a survival time, which is the variable of interest. In this paper, we apply the martingale method for counting processes to study asymptotic properties for the kernel estimator of the density function of the survival times. We also derive an asymptotic expression for the mean integrated square error of the kernel density estimator, which can be used to obtain an asymptotically optimal bandwidth.