温州大学数理学院2025年第6期数理大讲堂

报告题目：Multivariate two-sample tests via random projection

报告时间： 2025年5月20日上午9:00

报告地点：3B205

主讲人：杜江，教授，博导， 北京工业大学

主持人：高利新 教授

摘要：

The two-sample test problem is a fundamental problem in statistical inference that attempts to test whether two probability measures are different based on corresponding samples. Consequently, many statistical methods have been proposed when random vectors are multivariate or even high-dimensional. For this problem, we introduce a randomly projected maximum mean discrepancy (MMD) in a reproducing kernel Hilbert space to characterize the distance between the distributions of two random vectors. The multivariate random vectors are projected onto univariate random variables and projected MMD statistics are constructed.

The collection of projected MMD indexed by the unit sphere, and hence we treat it as the $U$-process. Theorems include the asymptotic theory of test statistics under the null hypothesis and the alternative hypothesis.

Combining continuous mapping theory with projected MMD statistics, a class of test statistics is proposed, which includes the Cram\'er-von Mises and Kolmogorov-Smirnov methods as special cases.

Since the limit null distribution of the test statistic depends on the data generation process, we apply the permutation test procedure to determine a critical value. Furthermore, the empirical size and power of the test statistics are evaluated by numerical simulations. Finally, we illustrate our method by applying it to real data sets with two-sample problem.

主讲人介绍：

杜江，北京工业大学，数学统计学与力学学院，教授，博士生导师。主持国家自然科学基金项目2项，中国博士后基金1项，北京市教委科技计划项目1项，参加国家重点研发计划2项、国家自然科学基金3项、国家社科科学基金2项。已在国内外学术刊物上发表论文50 余篇，其中40余篇被SCI检索。研究方向为空间数据分析、模型检验、高维数据分析等。