On the asymptotic power of a method for testing hypothesis about equality of distributions

The paper is devoted to studying the asymptotic power of a method for testing hypothesis on equality of two distributions that can be considered as a generalization of Mann - Whitney -Wilcoxon test. We consider a class of distributions such that the expection of the square of an auxiliary function is finite. For the case when alternative distribution differ from the initial one only by a shift the asymptotic distribution and asymptotic power of the test are found explicitly. Up to now the power of the test was studied only by stochastic simulation.