On asymptotic minimaxity of Kolmogorov and omega-square tests

We consider the problem of hypothesis testing about a value of functional. For a given functional T the problem is to test a hypothesis T(P) = 0 versus alternatives T(P) > b0 > 0 where P is an arbitrary probability measure. Under the natural assumptions we show that the test statistics T(P̂n) depending on the empirical probability measures P̂n are asymptotically minimax. Since the sets of alternatives is fixed the asymptotic minimaxity is considered in the senses of Bahadur and Hodges-Lehmann efficiencies. In particular the functional T can be the functional corresponded to the test statistics of Kolmogorov and omega-square tests