Sequential testing of two composite statistical hypotheses

Consideration was given to the sequential testing of two composite statistical hypotheses. Each hypothesis is described by the density depending on the parameter which can belong to any of the two disjoint sets. A sequential procedure minimizing the Bayes risk which is maximal in the family of the a priori parameter distributions was proposed. The family of a priori distributions consists of all probabilistic distributions over the parametric set for which the a priori probability of validity of one of the composite hypotheses is equal to the given value. It was established that this procedure minimizes the parameter-maximal mean observation time under an additional condition and with the assumption of validity of any of the hypotheses in the class of rules for which the parameter-maximal error probabilities do not exceed the given values. The results obtained change over to the Wald classical results for the case of testing simple hypotheses.