Multivariate Two-Sample Tests Via Random Projection

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 Cramér-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.