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 $U$.
Citation
Roger A. Shorack. "Recursive Generation of the Distribution of the Mann-Whitney-Wilcoxon U-Statistic Under Generalized Lehmann Alternatives." Ann. Math. Statist. 37 (1) 284 - 286, February, 1966. https://doi.org/10.1214/aoms/1177699621
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