Search

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 m n / (m + n) -consistent under the strong null and local alternatives, and root- m n / (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. [ABSTRACT FROM AUTHOR]

Testing for homogeneity of two random vectors is a fundamental problem in statistics. In the past two decades, numerous efforts have been made to detect heterogeneity when the random vectors are multivariate or even high dimensional. Due to the "curse of dimensionality", existing tests based on Euclidean distance may fail to capture the overall homogeneity in high-dimensional settings while can only capture the moment discrepancy. To address this issue, we propose a fully nonparametric test for homogeneity of two random vectors. Our method involves randomly selecting two subspaces consisting of components of the vectors, projecting the subspaces onto one-dimensional spaces, respectively, and constructing the test statistic using the Cramér–von Mises distance of the projections. To enhance the performance, we repeatedly implement this procedure to construct the final test statistic. Theoretically, if the replication time tends to infinity, we can avoid potential power loss caused by lousy directions. Owing to the U -statistic theory, the asymptotic null distribution of our proposed test is standard normal, regardless of the parent distributions of the random samples and the relationship between data dimensions and sample sizes. As a result, no re-sampling procedure is needed to determine critical values. The empirical size and power of the proposed test are demonstrated through numerical simulations. [ABSTRACT FROM AUTHOR]