Abstract
Due to the lack of a canonical ordering in for , 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 (Ann. Statist. 45 (2017) 223–256; Hallin (2017); Ann. Statist. 49 (2021) 1139–1165), thereby inducing ranks, signs, and a data-driven ordering of . 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.
Funding Statement
Hongjian Shi and Mathias Drton were supported by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No 883818). Marc Hallin acknowledges the support of the Czech Science Foundation grant GAČR22036365. Fang Han was supported by the United States NSF grants DMS-1712536 and SES-2019363.
Acknowledgements
The authors would like to thank two anonymous referees, an anonymous Associate Editor and the Editor Davy Paindaveine for their constructive comments that improved the quality of this paper.
Citation
Hongjian Shi. Mathias Drton. Marc Hallin. Fang Han. "Distribution-free tests of multivariate independence based on center-outward quadrant, Spearman, Kendall, and van der Waerden statistics." Bernoulli 31 (1) 106 - 129, February 2025. https://doi.org/10.3150/24-BEJ1721
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