Distribution-free tests of multivariate independence based on center-outward quadrant, Spearman, Kendall, and van der Waerden statistics

TL;DR

This paper introduces distribution-free multivariate independence tests based on optimal transport ranks, extending classical univariate rank-based tests to higher dimensions with asymptotic theory and efficiency analysis.

Contribution

It develops new multivariate independence tests using optimal transport-based ranks, providing their asymptotic distribution, power analysis, and efficiency comparisons with classical tests.

Findings

Tests are fully distribution-free and valid for any absolutely continuous distribution.

Asymptotic distribution theory and critical value approximations are established.

The tests demonstrate favorable efficiency properties, including a Chernoff--Savage type result.

Abstract

Due to the lack of a canonical ordering in ${\mathbb R}^d$ for $d>1$, defining multivariate generalizations of the classical univariate ranks has been a long-standing open problem in statistics. Optimal transport has been shown to offer a solution in which multivariate ranks are obtained by transporting data points to a grid that approximates a uniform reference measure (Chernozhukov et al., 2017; Hallin, 2017; Hallin et al., 2021), thereby inducing ranks, signs, and a data-driven ordering of ${\mathbb R}^d$. We take up this new perspective to define and study multivariate analogues of the sign covariance/quadrant statistic, Spearman's rho, Kendall's tau, and van der Waerden covariances. The resulting tests of multivariate independence are fully distribution-free, hence uniformly valid irrespective of the actual (absolutely continuous) distribution of the observations. Our results provide the asymptotic distribution theory for these new test statistics, with asymptotic approximations to critical values to be used for testing independence between random vectors, as well as a power analysis of the resulting tests in an extension of the so-called (bivariate) Konijn model. This power analysis includes a multivariate Chernoff--Savage property guaranteeing that, under elliptical generalized Konijn models, the asymptotic relative efficiency of our van der Waerden tests with respect to Wilks' classical (pseudo-)Gaussian procedure is strictly larger than or equal to one, where equality is achieved under Gaussian distributions only. We similarly provide a lower bound for the asymptotic relative efficiency of our Spearman procedure with respect to Wilks' test, thus extending the classical result by Hodges and Lehmann on the asymptotic relative efficiency, in univariate location models, of Wilcoxon tests with respect to the Student ones.