On Stable Rules for Discrimination of Hypotheses

The article considers the problem of discriminating several hypotheses, each of which comprises a family of densities bounded on both sides. The comparison criterion is the maximum with respect to the unknown densities of the average risk. Rules are obtained for minimizing the maximum risk. Under certain conditions, these rules may be based on both the upper and the lower bounds of the families of densities. Since, in accordance with the maximum-likelihood method, rules are based only on upper bounds for families of densities, in some sense the rules can be both optimal and nonoptimal.