Sequential Compound Rules for the Finite Decision Problem

Summary Asymptotically optimal rules are obtained for the sequential compound decision problem, where the component problem involves a finite parameter space and a finite action space. It is shown that under some regularity conditions the rule, which at each stage is a Bayes rule with respect to an estimate of the empirical distribution of the previous parameters, is asymptotically optimal. In the general situation, a randomized version of this rule is asymptotically optimal. Convergence of the compound risks and of the corresponding losses are considered.