## The Annals of Statistics

### Bayesian Nonparametric Estimation Based on Censored Data

#### Abstract

Let $X_1, \cdots, X_n$ be a random sample from an unknown $\operatorname{cdf} F$, let $y_1, \cdots, y_n$ be known real constants, and let $Z_i = \min(X_i, y_i), i = 1, \cdots, n$. It is required to estimate $F$ on the basis of the observations $Z_1, \cdots, Z_n$, 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.

#### Article information

Source
Ann. Statist., Volume 7, Number 1 (1979), 163-186.

Dates
First available in Project Euclid: 12 April 2007

https://projecteuclid.org/euclid.aos/1176344562

Digital Object Identifier
doi:10.1214/aos/1176344562

Mathematical Reviews number (MathSciNet)
MR515691

Zentralblatt MATH identifier
0401.62031

JSTOR