Two-sample distribution tests in high dimensions via max-sliced Wasserstein distance and bootstrapping

SummaryTwo-sample hypothesis testing is a fundamental statistical problem for inference about two populations. In this paper, we construct a novel test statistic to detect high-dimensional distributional differences based on the max-sliced Wasserstein distance to mitigate the curse of dimensionality. By exploiting an intriguing link between the distance and suprema of empirical processes, we develop an effective bootstrapping procedure to approximate the null distribution of the test statistic. One distinctive feature of the proposed test is the ability to construct simultaneous confidence intervals for the max-sliced Wasserstein distances of projected distributions of interest. This enables, not only the detection of global distributional differences, but also the identification of significantly different marginal distributions between two populations, without the need for additional tests. We establish the convergence of Gaussian and bootstrap approximations of the proposed test, based on which we show that the test is asymptotically valid and powerful as long as the considered max-sliced Wasserstein distance is adequately large. The merits of our approach are illustrated via simulated and real data examples.