Optimal Sequential Tests for Testing Two Composite and Multiple Simple Hypotheses

This article considers the problem of sequential testing of two composite hypotheses. Each of the hypotheses is described by a probability density function depending on a parameter. The parameter can belong to one of the two disjoint subsets of a given set. We present a sequential procedure that minimizes the Bayesian risk maximal over a family of prior parameter distributions. The family of prior distributions consists of all of the probabilistic distributions on the parametric set such that the prior probability of one of the hypotheses is equal to a given number. Consider the family of all of the possible sequential decision rules with given constraints on the greatest, with respect to the parameter, error probabilities. We prove that the procedure minimizes the maximal value (with respect to the parameter) of the average run length assuming the validity of any of the above two hypotheses over the family. In the case of two simple hypotheses sequential testing problems, our results give rise to the classical Wald-Wolfowitz theorem. Furthermore, the problem of sequential testing of three, or more, simple hypotheses becomes a special case of our results as well.