Multivariate Symmetry: Distribution-Free Testing via Optimal Transport

TL;DR

This paper introduces a new framework for distribution-free multivariate symmetry testing using optimal transport, extending classical nonparametric tests like sign and Wilcoxon signed-rank to higher dimensions.

Contribution

It develops a unified approach for multivariate symmetry testing based on optimal transport, creating distribution-free signs, ranks, and signed-ranks with strong theoretical properties.

Findings

Tests are exactly distribution-free in finite samples.

Proposed tests are asymptotically normal and adapt to various symmetry notions.

GWSR test is highly powerful against location shifts, matching parametric efficiency.

Abstract

The sign test (Arbuthnott, 1710) and the Wilcoxon signed-rank test (Wilcoxon, 1945) are among the first examples of a nonparametric test. These procedures -- based on signs, (absolute) ranks and signed-ranks -- yield distribution-free tests for symmetry in one-dimension. In this paper we propose a novel and unified framework for distribution-free testing of 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, 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 $T^2$ 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.