Abstract: Many tests are available to test the 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. This limits their usefulness in high dimensions, even when the sample sizes are sufficiently large. In addition, existing tests require a careful selection of the tuning parameters to enhance their power performance. However, doing so is very challenging, especially in high dimensions. In this paper, we propose a robust and fully nonparametric two-sample test to detect the heterogeneity of two random samples. Our proposed test is 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 that no bootstrap or re-sampling technique has to be used to decide an appropriate critical value. We demonstrate the power performance of our proposed test through extensive simulations and real-world applications.

Key words and phrases: Cramér-von Mises test, equality of distributions, high dimension, homogeneity, two-sample test, U-statistics.