A Generalized Wilcoxon-Mann-Whitney Statistic With Utility For Interpoint Distance Analysis

We develop a simple but useful generalization of the classical Wilcoxon-MannWhitney statistic. A normal approximation and a recurrence for the exact distribution of this generalization are available. The statistic has potential application in nonparametric discriminant analysis. 1. INTRODUCTION Consider a nonparametric rank-based test for location in the two-sample case. Let be i.i.d. and be i.i.d. with , mutually independent. We wish to test against the alternative of stochastic ordering. We will assume for simplicity that the distributions are continuous, implying that rank ties occur with probability zero. The Wilcoxon-Mann-Whitney (WMW) statistic (Wilcoxon 1945; Mann and Whitney 1947) is based on pairwise comparisons; . The statistic is an estimator of ; and the associated test rejects for large values of . We present a generalized WMW statistic based on the comparison of subsample minima. Let and , and define c 1 X 1 1 ... X n 1 1 , , { } = F 1 c 2 X 1 2 ... X n 2 ..