Multivariate Rank-based Distribution-free Nonparametric Testing using Measure Transportation

TL;DR

This paper introduces a measure transportation-based framework for multivariate distribution-free nonparametric tests, enabling exact, tuning-free, and computationally feasible testing procedures for independence and distribution equality.

Contribution

It develops a novel multivariate rank concept using measure transportation, leading to new distribution-free tests with proven asymptotic properties and broad applicability.

Findings

Tests are exactly distribution-free under the null hypothesis.

Proposed tests are consistent against all fixed alternatives.

Extensive simulations demonstrate the effectiveness of the methods.

Abstract

In this paper, we propose a general framework for distribution-free nonparametric testing in multi-dimensions, based on a notion of multivariate ranks defined using the theory of measure transportation. Unlike other existing proposals in the literature, these multivariate ranks share a number of useful properties with the usual one-dimensional ranks; most importantly, these ranks are distribution-free. This crucial observation allows us to design nonparametric tests that are exactly distribution-free under the null hypothesis. We demonstrate the applicability of this approach by constructing exact distribution-free tests for two classical nonparametric problems: (i) testing for mutual independence between random vectors, and (ii) testing for the equality of multivariate distributions. In particular, we propose (multivariate) rank versions of distance covariance (Sz\'ekely et al., 2007) and energy statistic (Sz\'ekely and Rizzo, 2013) for testing scenarios (i) and (ii) respectively. In both these problems, we derive the asymptotic null distribution of the proposed test statistics. We further show that our tests are consistent against all fixed alternatives. Moreover, the proposed tests are tuning-free, computationally feasible and are well-defined under minimal assumptions on the underlying distributions (e.g., they do not need any moment assumptions). We also demonstrate the efficacy of these procedures via extensive simulations. In the process of analyzing the theoretical properties of our procedures, we end up proving some new results in the theory of measure transportation and in the limit theory of permutation statistics using Stein's method for exchangeable pairs, which may be of independent interest.