On UMPS hypothesis testing

Abstract

For two-sided hypothesis testing in location families, the classical optimality criterion is the one leading to uniformly most powerful unbiased (UMPU) tests. Such optimal tests, however, are constructed in exponential models only. We argue that if the base distribution is symmetric, then it is natural to consider uniformly most powerful symmetric (UMPS) tests, that is, tests that are uniformly most powerful in the class of level-\(\alpha \) tests whose power function is symmetric. For single-observation models, we provide a condition ensuring existence of UMPS tests and give their explicit form. When this condition is not met, UMPS tests may fail to exist and we provide a weaker condition under which there exist UMP tests in the class of level-\(\alpha \) tests whose power function is symmetric and U-shaped. In the multi-observation case, we obtain results in exponential models that also allow for non-location families.