A Study of a Class of Weighted Bootstraps for Censored Data

Edgeworth expansions are derived for a class of weighted bootstrap methods for the Kaplan-Meier and Nelson-Aalen estimates using the methods contained in the monograph by Barbe and Bertail (1995). Von Mises representations up to the third order are established for the weighted bootstrap versions of these estimators. It is shown that there exists weights which out-perform Efron's bootstrap method in terms of coverage accuracy. Moreover, it is shown that this holds for a particular choice of gamma weights which are very easy to use in practice. The general weighting schemes are also useful in approximating the posterior distribution of a survival function with respect to mixtures of beta-neutral process priors. 1. Introduction Lo (1993a) introduced the idea of a censored data Bayesian bootstrap. This is the direct analogue to Rubin's (1981) Bayesian bootstrap (BB) in the complete data case and provides an alternative to Efron's (1981) bootstrap method for the Kaplan-Meier estimator. James (1993) studied a generalization of this idea to the case of arbitrary i.i.d. weights. This proves the consistency result , under a general random censoring model, for these class of weights using martingale techniques in the spirit of Gill (1980). From the work of Mason and Newton (1992) see also Praestgaard and Wellner (1993)] it is perhaps not diicult to see that one may further deene the notion of a generalized exchangeably weighted Kaplan-Meier bootstrap estimate, we will mention this brieey in the coming paragraphs. Perhaps a more interesting question, and the main focus of this paper, is to determine the proper choice of weights for a class of weighted bootstrapped Kaplan-Meier estimators and related functionals. The recent work of Barbe and Bertail (1995) involving Edgeworth expansions for a class of weighted bootstrap versions of general von Mises diierentiable functionals provides the basis for this study. Prior to their work, most of the results on Edgeworth expansions for the weighted bootstrap has been relegated to the case of the sample mean. Related works include