On empirical bayes rules for selecting the best hypergeometric population

In this paper, we consider k hypergeometric populations πi = π(Mi,mi,si), i=1,…,k, where Mi is the number of units in population πi, mi is the number of units selected from πi, and si is the number of defective units in πi . Let . A population πi with is considered as a best population. We are interested in selecting the best population and the best population compared with a control. It is assumed that the unknown parameteres si i = 1,…,k, follow some binomial prior distribution with unknown hyperparameters. Under the hypergeometric-binomial model, two empirical Bayes selection rules are studied according to the different selection problems. It is shown that for each empirical Bayes selection rule, the corresponding Bayes risk tends to the minimum Bayes risk with a rate of convergence of order O(exp(-cn)) for some positive constant c, where the value of c varies depending on the rule and n is the number of accumulated past observations.