SINGLE-SAMPLE BAYES AND EMPIRICAL BAYES RULES FOR RANKING AND ESTIMATING MULTINOMIAL PROBABILITIES

The problem of selecting the most probable cell, more generally the problem of partitioning k multinomial cells according to the values of the cell probabilities is considered. Bayes rules are obtained for the general ranking problem under a general class of loss functions. A sequence of parametric empirical Bayes selection rules is proposed and shown to be asymptotically optimal of order O(e(-cn)) for some positive constant c. Also the problem of selecting the most (least) probable cell and simultaneously estimating the associated probability of the selected cell is considered. For this problem also a sequence of parametric empirical Bayes rules is proposed and shown to be asymptotically optimal of order O(n-1).