ELSEVIER Journal of Statistical Planning and Inference 69 (1998) 115-131 joumal of statistical planning and inference Bootstrapping quantiles in a fixed design regression model with censored data Ingrid Van Keilegom, NoEl Veraverbeke * Limburgs Universitair Centrum, Universitaire Campus, B-3590 Diepenbeek, Belgium Received 18 March 1996; received in revised form 2 July 1997 Abstract We consider the problem of estimating the quantiles of a distribution function in a fixed design regression model in which the observations are subject to random right censoring. The quantile estimator is defined via a conditional Kaplan-Meier type estimator for the distribution at a given design point. We establish an a.s. asymptotic representation for this quantile estimator, from which we obtain its asymptotic normality. Because a complicated estimation procedure is necessary for estimating the asymptotic bias and variance, we use a resampling procedure, which provides us, via an asymptotic representation for the bootstrapped estimator, with an alternative for the normal approximation. @ 1998 Elsevier Science B.V. All rights reserved. A M S classifications: Primary 62G09; secondary 62G05, 62G20 Keywords: Asymptotic representation; Bootstrap; Fixed design; Nonparametric regression; Quantiles; Right censoring 1. Introduction We consider a regression model with fixed design points 0 ~<xl ~< • • • ~<x, ~< 1. These design points represent, e.g. the dose of a drug that n individuals in a clinical study receive. At these points xi, we observe nonnegative and independent responses Yi, rep- resenting, e.g. the survival times of the individuals in the study. We denote their distri- bution function (d.f.) by Fx,(t)= P(Y/<~ t). The responses Yi (i = 1..... n) are subject to random right censoring, i.e. instead of observing Y/, we only observe Ti and 6i, where Ti =min(Yi, Ci) and 6i =I(Yi<~Ci) and where the random variables (r.v.) Ci are non- negative and independent censoring variables, representing, e.g. withdrawal time from the study or time until death from a cause unrelated to the study. Further, we use the notations Gx,(t)=P(Ci <~t) and Hx,(t)=P(Ti <~t) and we assume that Y/ and Ci are in- dependent for each i. This entails that 1-Hx, = (1-Fx, )( 1- Gxi ). At a given design point * E-mail: nveraver@luc.ac.be. 0378-3758/98/$19.00 (~) 1998 Elsevier Science B.V. All rights reserved. PH S0378-3758(97)001 26-2