ESTIMATION AND BOOTSTRAP WITH CENSORED DATA IN FIXED DESIGN NONPARAMETRIC REGRESSION

INGRID VAN KEILEGOM AND NOËL VERAVERBEKE

Department of Mathematics, Limburgs Universitair Centrum, Universitaire Campus, B-3590 Diepenbeek, Belgium

(Received March 1, 1996; revised June 18, 1996)

Abstract. We study Beran’s extension of the Kaplan-Meier estimator for the situation of right censored observations at fixed covariate values. This estimator for the conditional distribution function at a given value of the covariate involves smoothing with Gasser-Müller weights. We establish an almost sure asymptotic representation which provides a key tool for obtaining central limit results. To avoid complicated estimation of asymptotic bias and variance parameters, we propose a resampling method which takes the covariate information into account. An asymptotic representation for the bootstrapped estimator is proved and the strong consistency of the bootstrap approximation to the conditional distribution function is obtained.

Key words and phrases: Asymptotic normality, asymptotic representation, bootstrap approximation, fixed design, kernel estimator, nonparametric regression, right censoring.