On the asymptotic optimality of the Bayesian decision rule in the problem of multiple classification of hypotheses

This paper considers the problem of constructing a k-element decision set for a totality of m hypotheses concerning the form of the polynomial distribution. In the paper of [Theory Probab. Appl., 43 (1998), pp. 239–255] upper and lower estimates were obtained for the asymptotics of the maximum of logarithms of probabilities of errors of the ith kind, i.e., the probabilities for the optimally constructed k-element decision set not containing the index $i$ under the condition that i is the number of the true hypothesis. In the present paper we establish the coincidence of these estimates and, thus, make the result of Salikhov somewhat complete. As this is so, it is implicitly assumed that the prior distribution on the totality of m hypotheses is uniform. Moreover, in passing, the asymptotic optimality of the Bayesian method of construction of the decision set is established.