Pitman efficiency lower bounds for multivariate distribution-free tests based on optimal transport

Abstract The Wilcoxon rank sum test is one of the most popular distribution-free two-sample tests for univariate data. Among the important reasons for their popularity are the striking results of Hodges–Lehmann and Chernoff–Savage, where the authors show that the asymptotic (Pitman) relative efficiency of Wilcoxon’s test compared to Student’s t-test, never falls below 0.864 (with identity score) and 1 (with Gaussian score), respectively. Motivated by these results, we propose and study a large family of exactly distribution-free multivariate rank-based two-sample tests by leveraging the theory of optimal transport. First, we propose distribution-free analogues of the Hotelling T2 test and show that they satisfy Hodges–Lehmann and Chernoff–Savage-type efficiency lower bounds over natural sub-families of multivariate distributions—making them the first multivariate, nonparametric, finite-sample distribution-free tests that provably achieve such efficiency lower bounds. Next, we propose exactly distribution-free versions of the celebrated kernel maximum mean discrepancy test. In addition to being distribution-free in finite-samples, these tests are universally consistent under no moment assumptions and have nontrivial Pitman efficiency. To the best of our knowledge, these are the first class of tests to have this trifecta of properties.