Most powerful conditional tests

The present paper establishes finite sample most powerful tests for certain nonparametric null hypotheses P₀ which admit a sufficient statistic S. The underlying alternatives are of semiparametric or nonparametric nature. Optimal one-sided S-conditional test are offered for families with nonparametric isotone likelihood ratio. Similarly two-sided optimal locally unbiased S-conditional test are introduced for alternatives with nonparametric convex likelihood. If in addition S is P₀-complete then of course we arrive at most powerful α-similar tests. Special examples are randomization tests, permutation tests for two-sample problems and symmetry tests for the null hypothesis of 0-symmetry. The results rely on a new conditional Neyman–Pearson Lemma which can be found in the appendix and which is of own interest. This Lemma is used to solve conditional optimization problems for tests.