The Annals of Statistics

Nonparametric hierarchical Bayes via sequential imputations

Jun S. Liu

Full-text: Open access

Abstract

We consider the empirical Bayes estimation of a distribution using binary data via the Dirichlet process. Let $\mathscr{D}(\alpha)$ denote a Dirichlet process with $\alpha$ being a finite measure on Instead of having direct samples from an unknown random distribution F from $\mathscr{D}(\alpha)$, we assume that only indirect binomial data are observable. This paper presents a new interpretation of Lo's formula, and thereby relates the predictive density of the observations based on a Dirichlet process model to likelihoods of much simpler models. As a consequence, the log-likelihood surface, as well as the maximum likelihood estimate of $c = \alpha([0, 1])$, is found when the shape of $\alpha$ a is assumed known, together with a formula for the Fisher information evaluated at the estimate. The sequential imputation method of Kong, Liu and Wong is recommended for overcoming computational difficulties commonly encountered in this area. The related approximation formulas are provided. An analysis of the tack data of Beckett and Diaconis, which motivated this study, is supplemented to illustrate our methods.

Article information

Source
Ann. Statist. Volume 24, Number 3 (1996), 911-930.

Dates
First available in Project Euclid: 20 September 2002

Permanent link to this document
http://projecteuclid.org/euclid.aos/1032526949

Mathematical Reviews number (MathSciNet)
MR1401830

Digital Object Identifier
doi:10.1214/aos/1032526949

Zentralblatt MATH identifier
0880.62038

Subjects
Primary: 62G05: Estimation
Secondary: 62E25 65U05

Keywords
Dirichlet process empirical Bayes Gibbs sampler importance sampling Pólya urn sensitivity analysis

Citation

Liu, Jun S. Nonparametric hierarchical Bayes via sequential imputations. Ann. Statist. 24 (1996), no. 3, 911--930. doi:10.1214/aos/1032526949. http://projecteuclid.org/euclid.aos/1032526949.


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