## The Annals of Statistics

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

### Bayesian Nonparametric Estimation Based on Censored Data

Thomas S. Ferguson and Eswar G. Phadia

#### 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

**Permanent link to this document**

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**

links.jstor.org

**Subjects**

Primary: 62C10: Bayesian problems; characterization of Bayes procedures

Secondary: 62G05: Estimation 60K99: None of the above, but in this section

**Keywords**

Bayesian nonparametric estimation survival function censored data prior distribution process neutral to the right Dirichlet process processes with independent increments modal estimation

#### Citation

Ferguson, Thomas S.; Phadia, Eswar G. Bayesian Nonparametric Estimation Based on Censored Data. Ann. Statist. 7 (1979), no. 1, 163--186. doi:10.1214/aos/1176344562. https://projecteuclid.org/euclid.aos/1176344562