Appears in collection : 2022 - T3 - WS3 - Measure-theoretic Approaches and Optimal Transportation in Statistics

We study the problem of testing symmetry in multidimensions. Multivariate generalizations of the sign and Wilcoxon signed rank test are proposed using the theory of optimal transport. The proposed tests are exactly distribution-free in finite samples, with an asymptotic normal distribution, and adapt to various notions of multivariate symmetry such as central symmetry, sign symmetry, and spherical symmetry. We study the consistency of the proposed tests and their behaviors under local alternatives, and show that in a large class of scenarios, our generalized Wilcoxon signed rank test suffers from no loss in asymptotic relative efficiency, when compared to the Hotelling's $T^2$ test, despite being nonparametric and exactly distribution-free.