LAUSR

Many tests are proposed in the literature to test homogeneity of two random samples, that is, the exact equivalence of their statistical distributions. When the two random samples are high-dimensional or not normally distributed, the asymptotic null distributions of most existing two-sample tests are rarely tractable, which limits their usefulness in high dimensions even when the sample sizes are sufficiently large. In addition, existing tests require to select tuning parameters delicately to enhance power performance. However, how to select optimal tunings is very challenging, especially in high dimensions. In this paper, we propose a robust and fully nonparametric two-sample test to detect heterogeneity of two random samples. Our proposed test is completely free of tuning parameters. It is built upon the Cramér-von Mises distance and can be readily used in high dimensions. In addition, our proposed test is robust to the presence of outliers or extreme values in that no moment condition is required. The asymptotic null distribution of our proposed test is standard normal, when both the sample sizes and the dimensions of the two random samples diverge to infinity. This facilitates the implementation of our proposed test dramatically, in Statistica Sinica: Newly accepted Paper (accepted version subject to English editing)