On the asymptotic power of the “energy” test for equality of two distributions

In the paper the asymptotic distribution and the formula for the asymptotic power are found for the “energy” test in the case of alternative distributions differ from zero distribution by the parameter of shift and/or the parameter of scale. This criterion is an alternative for the well-known Mann-Whitney test but allows to compare distributions that differ by the scale parameter. The efficiency of the results obtained is demonstrated by the stochastic simulation for the normal distribution and the Cauchy distribution.