*(English)*Zbl 0880.62038

Summary: We consider the empirical Bayes estimation of a distribution using binary data via the Dirichlet process. Let $\mathcal{D}\left(\alpha \right)$ 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 $\mathcal{D}\left(\alpha \right)$, we assume that only indirect binomial data are observable.

This paper presents a new interpretation of Lo’s formula [*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 $c=\alpha \left(\right[0,1\left]\right)$, 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 *A. Kong, J. S. Liu* and *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 *L. Beckett* and *P. Diaconis* [Adv. Math. 103, No. 1, 107-128 (1994; Zbl 0805.62085)] which motivated this study, is supplemented to illustrate our methods.

##### MSC:

62G05 | Nonparametric estimation |

62C12 | Empirical decision procedures; empirical Bayes procedures |

65C05 | Monte Carlo methods |

65C99 | Probabilistic methods, simulation and stochastic differential equations (numerical analysis) |