A generalized Wilcoxon–Mann–Whitney type test for multivariate data through pairwise distance

The Wilcoxon–Mann–Whitney test is designed to test homogeneity of two random samples in the univariate case. It is very powerful to detect location shifts yet may lose power completely when there exist scale differences. We generalize the classic Wilcoxon–Mann–Whitney test through using pairwise distances of all observations. The generalized test can be readily used even when the random observations are multivariate. It is also very powerful in the presence of scale differences. The generalized test is in spirit to compare difference between the distribution functions of two random samples. It is mn/(m+n)-consistent under the strong null and local alternatives, and root-mn/(m+n)-consistent under fixed alternatives, where m,n stand for the respective sizes of the two random samples. The power of the generalized test is asymptotically independent of m/n, the size ratio of the two random samples. This indicates that the generalized test has nontrivial power as long as the sample sizes are not extremely unbalanced. We demonstrate the theoretical properties of our improved rank-based two-sample test through comprehensive numerical studies.