Empirical Bayes rules for selecting the most and least probable multivariate hypergeometric event

In this paper, we consider a multivariate hypergeometric population with k-category [pi] = [pi](N, M, s1,...,sk), where si, I = 1,...,k, is the number of units in category [pi]i, with [Sigma]i-1k si = M, the total number of units in population [pi], and N is the number of units selected from population [pi]. Let s[1] = min1[less-than-or-equals, slant]i[less-than-or-equals, slant]k si, s[k] = max1[less-than-or-equals, slant]i[less-than-or-equals, slant]k si. A category associated with si = s[k] (or si = s[1]) is considered as the most (or the least) probable event. We are interested in selecting the most and the least probable event. It is assumed that the unknown parameters si, I = 1,...,k, follow a multinomial prior distribution with unknown hyperparameters. Under this statistical framework, two empirical Bayes selection rules are studied according to the two 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 at hand.