报告题目：Robust nonparametric two-sample tests for equality of distributions in high dimensions

报告人：朱利平 教授 中国人民大学

报告时间：2018年8月30日下午2:40-3:40

报告地点：数学楼第一报告厅

报告摘要：We propose a robust nonparametric two-sample test, which generalizes the Cram\'er-von Mises test through projections, to test for equality of two distributions in high dimensions. The population version of our proposed generalized Cram\'er-von Mises statistic is nonnegative and equals zero if and only if the two distributions are identical, ensuring that our proposed test is consistent against all fixed alternatives. In addition, our proposed test statistic has an explicit form and is completely free of tuning parameters. It requires no moment conditions and hence is robust to the presence of outliers and heavy-tail observations. We study the asymptotic behaviors of our proposed test under both the ``large sample size, fixed dimension" and the ``fixed sample size, large dimension" paradigms. In the former paradigm, we show that the asymptotic power of our proposed test does not depend on the size ratio of the two random samples. This ensures that our proposed test can be readily applied to imbalanced samples. In the latter paradigm, we observe that, surprisingly, the two distributions are equal if and only if their first two moments are equal. Therefore, we suggest to tailor our proposed test to detect location shifts and scale differences, which further enhances the power performance of our proposed test significantly. Numerical studies confirm that our proposals are superior to many existing tests in high dimensional two-sample test problems.