Maximal asymptotic power and efficiency of two-sample tests based on generalized U-Statistics

In this article a systematic study is given of the asymptotic behavior of two-sample tests based on U-Statistics with arbitrary antisymmetric kernels psi. Besides the investigation under the hypothesis and under fixed alternatives we determine the local power as a function of psi as well as its maximizing value psi(opt). Moreover formulas for the asymptotic relative efficiency ARE(psi(2),psi(1)) of the psi(2)-test with respect to the psi(1)-test are derived. It turns out that psi(opt) also yields the most efficient test in the sense that ARE(psi(opt),psi)less than or equal to1 for all (admissible) kernels psi.