Bayesian Nonparametric Estimation Based on Censored Data

Abstract

Let X1, ⋯, Xn be a random sample from an unknown $\operatorname{cdf} F$, let y1, ⋯, yn be known real constants, and let Zi = min(Xi, yi), i = 1, ⋯, n. It is required to estimate F on the basis of the observations Z1, ⋯, Zn, when the loss is squared error. We find a Bayes estimate of F when the prior distribution of F is a process neutral to the right. This generalizes results of Susarla and Van Ryzin who use a Dirichlet process prior. Two types of censoring are introduced--the inclusive and exclusive types--and the class of maximum likelihood estimates which thus generalize the product limit estimate of Kaplan and Meier is exhibited. The modal estimate of F for a Dirichlet process prior is found and related to work of Ramsey. In closing, an example illustrating the techniques is given.