On Hierarchical Bayesian Estimation and Selection for Multivariate Hypergeometric Distributions

Abstract

In this paper, we deal with the problem of simultaneous estimation for n independent multivariate hypergeometric distributions π(M _i ,m _i ,s _i ), i = 1,…,n, and simultaneous selection of the most probable event for each of the n π(M _i , m _i , s _i ). In order to model the uncertainty of the unknown parameters s _i , i = 1,…,n, and to incorporate information from the n π(M _i , m _i , s _i ), a two-stage prior distribution on the parameters s _i , i = 1,…,n, is introduced. With this hierarchical structure of prior distributions, we derive simultaneous hierarchical Bayesian estimators for s _i , i = 1,…,n, and simultaneous selection rule for selecting the most probable event for each of the n π(M _i , m _i , s _i ). We compare the performance of the hierarchical Bayesian procedures to those of the “pure” Bayesian procedures. The relative regret Bayes risks of the hierarchical Bayesian procedures are used as measures of the optimality of the procedures. It is shown that, for the estimation problem, the relative regret Bayes risk of the hierarchical Bayesian estimator converges to zero at a rate of order 0(n ^-1); and for the selection problem, the relative regret Bayes risk of the hierarchical Bayesian selection rule converges to zero at a rate of order 0(e ^-nβ ) for some positive constant β