A data-adaptive methodology for finding an optimal weighted generalized Mann–Whitney–Wilcoxon statistic

Xie and Priebe [2002. “Generalizing the Mann–Whitney–Wilcoxon Statistic”. J. Nonparametric Statist. 12, 661–682] introduced the class of weighted generalized Mann–Whitney–Wilcoxon (WGMWW) statistics which contained as special cases the classical Mann–Whitney test statistic and many other nonparametric distribution-free test statistics commonly used for the two-sample testing problem. The two-sample test that they proposed was based on any statistic within the class of WGMWW statistics optimal in the Pitman asymptotic efficacy (PAE) sense. In this paper, among other things, we show via simulation studies that for finite samples the PAE-optimal WGMWW test has substantially higher empirical power compared to the classical Mann–Whitney test for various underlying densities (especially for those densities for which Mann–Whitney test is considered a better alternative to parametric tests such as t  -tests). The PAE-optimal WGMWW test is not a candidate for the practitioner's toolbox since the corresponding test statistic contains parameters which are functions of the underlying null distribution function of the samples. The main thrust of this paper is in introducing a data-adaptive alternative to the PAE-optimal WGMWW test, which has efficacy and power as good as the latter. We provide an estimate ψ^ for the PAE function ψψ of a WGMWW statistic, and our test is based on a ψ^-optimal WGMWW statistic. We prove strong consistency of ψ^, thereby showing that our test has approximately the same efficacy as the ψψ-optimal WGMWW test for large sample sizes. Via simulation studies we show that for finite samples the empirical power of ψ^-optimal WGMWW test is almost the same as ψψ-optimal WGMWW test for various underlying densities. We also analyze magnetic imaging data related to subjects with and without Alzheimer's disease to illustrate our methodology. In summary, we present a strong competitor for the classical Mann–Whitney–Wilcoxon test and many other existing nonparametric distribution-free tests, especially for moderate and large samples.