Bayesian Nonparametric Estimation Based on Censored Data

Let XI, * * *, Xn be a random sample from an unknown cdf F, let Y 1. * , yn be known real constants, and let Z4 = min(Xi,yi), i = 1, * * *, n. It is required to estimate F on the basis of the observations ZI, , 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.