Distribution-Free Signs and Ranks via Optimal Transport under Multivariate Symmetry and Application to One-Sample Location Testing

We propose a novel and unified framework for distribution-free testing under multivariate symmetry (that includes central symmetry, sign symmetry, spherical symmetry, etc.) based on the theory of optimal transport. Our approach leads to notions of distribution-free generalized multivariate signs, absolute ranks and signed-ranks. As a consequence, we develop analogues of the sign and Wilcoxon signed-rank tests that share many of the appealing properties of their one-dimensional counterparts. In particular, the proposed tests are exactly distribution-free in finite samples with an asymptotic normal limit, and adapt to various notions of multivariate symmetry. We study the consistency of the proposed tests and their behavior under local alternatives, and show that the proposed generalized Wilcoxon signed-rank (GWSR) test is particularly powerful against location shift alternatives. We show that in a large class of such models, our GWSR test suffers from no loss in (asymptotic) efficiency, when compared to Hotelling’s T2 test, despite being nonparametric and exactly distribution-free. An appropriately score transformed version of the GWSR statistic leads to a locally asymptotically optimal test. Further, our method can be readily used to construct distribution-free confidence sets for the center of symmetry.