Center-Outward Sign- and Rank-Based Quadrant, Spearman, and Kendall Tests for Multivariate Independence

Marc Hallin, Hongjian Shi, Mathias Drton and Fang Han

No 2021-27, Working Papers ECARES from ULB -- Universite Libre de Bruxelles

Abstract: Defining multivariate generalizations of the classical univariate ranks has been a long-standing open problem in statistics. Optimal transport has been showed to offer a solution by transporting data points to grid approximating a reference measure (Chernozhukov et al. 2017;Hallin, 2017; Hallin et al. 2021a). We take up this new perspective to develop and study multivariate analogues of popular correlations measures including the sign covariance, Kendall's tau and Spearman's rho. Our tests are genuinely distribution-free, hence valid irrespective of the actual (absolutely continuous) distributions of the observations. We present asymptotic distribution theory for these new statistics, providing asymptotic approximations to critical values to be used for testing independence as well as an analysis of power of the resulting tests. Interestingly, we are able to establish a multivariate elliptical Chernoff-Savage property, which guarantees that, under ellipticity, our nonparametric tests of independence when compared to Gaussian procedures enjoy an asymptotic relative efficiency of one or larger. Hence, the nonparametric tests constitute a safe replacement for procedures based on multivariate Gaussianity.

Keywords: distribution-freeness; vector independence; rank tests; multivariate ranks (search for similar items in EconPapers)
Pages: 23 p.
Date: 2021-12
New Economics Papers: this item is included in nep-ecm
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