Given subsets $\Omega,\Phi$ of a set of probability measures, questions about the uniform in $P\in\Phi$ convergence of the normalized large deviations $n^{-1}\log P$ ($\hat P_n\in\Omega$) and about the convergence of the supremum over $\Phi$ of this value are considered for empirical distributions $\hat P_n$. The results are used for the proof of the asymptotic minimaxity of the Kolmogorov, omega-square, and rank tests by nonparametric sets of alternatives. A new bound for the efficiency of statistical tests is obtained. Bibliography: 19 titles.