HAZARD RATE ESTIMATION IN NONPARAMETRIC REGRESSION WITH CENSORED DATA

INGRID VAN KEILEGOM^{1} AND NOËL VERAVERBEKE^{2}

1Institut de Statistique, Université catholique de Louvain, Voie du Roman Pays 20, B-1348 Louvain-la-Neuve, Belgium

2Department of Mathematics, Limburgs Universitair Centrum, Universitaire Campus, B-3590 Diepenbeek, Belgium

(Received March 21, 2000; revised September 25, 2000)

Abstract. Consider a regression model in which the responses are subject to random right censoring. In this model, Beran studied the nonparametric estimation of the conditional cumulative hazard function and the corresponding cumulative distribution function. The main idea is to use smoothing in the covariates. Here we study asymptotic properties of the corresponding hazard function estimator obtained by convolution smoothing of Beran’s cumulative hazard estimator. We establish asymptotic expressions for the bias and the variance of the estimator, which together with an asymptotic representation lead to a weak convergence result. Also, the uniform strong consistency of the estimator is obtained.

Key words and phrases: Asymptotic representation, hazard rate, nonparametric regression, right censoring, weak convergence.