Asymptotic minimaxity of tests of the Kolmogorov and omega-square types

The asymptotic minimaxity of tests of the Kolmogorov and omega-square types is proved under natural nonparametric sets of alternatives. The tests are compared using a new asymptotic bound for the probabilities of moderate large deviations for the type I and type II errors. Our proof relies heavily on qualitatively new theorems about the probabilities of moderate large deviations of empirical measures.