Where variance is largest for Mann and Whitney's U under a sum of powers marginal distribution

This note is the proof of an important result for the application of random cut-points theory to the Mann-Whitney U test statistic, related directly to the Wilcoxon rank-sum statistic, used when data consist of frequencies in, for example, a 2 x k contingency table where the columns correspond to k ordered categories. The result is that if the cumulative probabilities of category assignment have a multi-parameter sum of powers e.d.f. on the unit interval, the variance is maximized where all the parameters are equal. Also, it is shown that the same result holds for the sum of mixed powers c.d.f. In these applications, the largest asymptotic variance corresponds to the largest asymptotic P-value for the test.