Recursive Generation of the Distribution of the Mann-Whitney-Wilcoxon U-Statistic Under Generalized Lehmann Alternatives

Abstract

Let $X$ and $Y$ be independent random variables having strictly increasing continuous distribution functions $F$ and $G$ respectively. It is shown that for hypotheses of the form $H : F = G$ versus $K_1 : G = F^k, k > 0$ or $H : F = G$ versus $K_2 : G = 1 - (1 - F)^k, k > 0$, the alternative distribution of the MWW $U$-statistic can be generated by a recursive formula analogous to the one used by Mann and Whitney in [2] to generate the null distribution of $U$.