On the structure of a maximin test in the problem of testing two composite hypotheses

We consider the problem of testing two composite hypotheses in the max-min setting, when we are looking for a randomized test that maximizes the minimum (over the alternative) of the power function in the class of all randomized tests of a given significance level. It is a part of folklore in mathematical statistics that a maximin test is a Neyman–Pearson test for a properly chosen Bayesian mixtures of measures corresponding to the hypothesis and to the alternative. We are interested in the case where there are no special assumptions on the hypothesis and the alternative; we only assume that a dominating measure exists.
In the talk we shall give a review of known results in this direction and present new results. We shall state three (dual) optimization problems, whose solutions allow us to characterize a maximin test. The existence of solutions to these dual problems is proved under assumptions that complement each other. Moreover, we can characterize a maximin test even if none of these assumptions is satisfied.
If time permits, we shall consider, in conclusion, similar optimization problems in case of finite measures, which is motivated by applications in mathematical finance.