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Nonparametric hierarchical Bayes via sequential imputations. (English) Zbl 0880.62038
Summary: We consider the empirical Bayes estimation of a distribution using binary data via the Dirichlet process. Let ${\cal D}(\alpha)$ denote a Dirichlet process with $\alpha$ being a finite measure on $[0,1]$. Instead of having direct samples from an unknown random distribution $F$ from ${\cal D}(\alpha)$, we assume that only indirect binomial data are observable. This paper presents a new interpretation of Lo’s formula [{\it A. Y. Lo}, ibid. 12, 351-357 (1984; Zbl 0557.62036)] 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 $\alpha$ is assumed known, together with a formula for the Fisher information evaluated at the estimate. The sequential imputation method of {\it A. Kong, J. S. Liu} and {\it W. H. Wong} [J. Am. Stat. Assoc. 89, No. 425, 278-288 (1994)] is recommended for overcoming computational difficulties commonly encountered in this area. The related approximation formulas are provided. An analysis of the tack data of {\it L. Beckett} and {\it P. Diaconis} [Adv. Math. 103, No. 1, 107-128 (1994; Zbl 0805.62085)] which motivated this study, is supplemented to illustrate our methods.

MSC:
62G05Nonparametric estimation
62C12Empirical decision procedures; empirical Bayes procedures
65C05Monte Carlo methods
65C99Probabilistic methods, simulation and stochastic differential equations (numerical analysis)
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References:
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