Semiparametrically Efficient Tests of Multivariate Independence Using Center-Outward Quadrant, Spearman, and Kendall Statistics

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). We take up this new perspective to develop and study multivariate analogues of the sign covariance/quadrant statistic, Kendall's tau, and Spearman's rho. The resulting tests of multivariate independence are genuinely distribution-free, hence uniformly valid irrespective of the actual (absolutely continuous) distributions of the observations. Our results provide asymptotic distribution theory for these new test statistics, with asymptotic approximations to critical values to be used for testing independence as well as a power analysis of the resulting tests. This includes a multivariate elliptical Chernoff-Savage property, which guarantees that, under ellipticity, our nonparametric tests of independence enjoy an asymptotic relative efficiency of one or larger with respect to the classical Gaussian procedures.