Bayesian Nonparametric Estimation for Incomplete Data Via Successive Substitution Sampling

Abstract

In the problem of estimating an unknown distribution function $F$ in the presence of censoring, one can use a nonparametric estimator such as the Kaplan-Meier estimator, or one can consider parametric modeling. There are many situations where physical reasons indicate that a certain parametric model holds approximately. In these cases a nonparametric estimator may be very inefficient relative to a parametric estimator. On the other hand, if the parametric model is only a crude approximation to the actual model, then the parametric estimator may perform poorly relative to the nonparametric estimator, and may even be inconsistent. The Bayesian paradigm provides a reasonable framework for this problem. In a Bayesian approach, one would try to put a prior distribution on $F$ that gives most of its mass to small neighborhoods of the entire parametric family. We show that certain priors based on the Dirichlet process prior can be used to accomplish this. For these priors the posterior distribution of $F$ given the censored data appears to be analytically intractable. We provide a method for approximating this posterior distribution through the use of a successive substitution sampling algorithm. We also show convergence of the algorithm.