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Distribution-Free Consistent Independence Tests via Center-Outward Ranks and Signs

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  • Hongjian Shi
  • Mathias Drton
  • Fang Han

Abstract

This article investigates the problem of testing independence of two random vectors of general dimensions. For this, we give for the first time a distribution-free consistent test. Our approach combines distance covariance with the center-outward ranks and signs developed by Marc Hallin and collaborators. In technical terms, the proposed test is consistent and distribution-free in the family of multivariate distributions with nonvanishing (Lebesgue) probability densities. Exploiting the (degenerate) U-statistic structure of the distance covariance and the combinatorial nature of Hallin’s center-outward ranks and signs, we are able to derive the limiting null distribution of our test statistic. The resulting asymptotic approximation is accurate already for moderate sample sizes and makes the test implementable without requiring permutation. The limiting distribution is derived via a more general result that gives a new type of combinatorial noncentral limit theorem for double- and multiple-indexed permutation statistics. Supplementary materials for this article are available online.

Suggested Citation

  • Hongjian Shi & Mathias Drton & Fang Han, 2022. "Distribution-Free Consistent Independence Tests via Center-Outward Ranks and Signs," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 117(537), pages 395-410, January.
  • Handle: RePEc:taf:jnlasa:v:117:y:2022:i:537:p:395-410
    DOI: 10.1080/01621459.2020.1782223
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    Cited by:

    1. S Gorsky & L Ma, 2022. "Multi-scale Fisher’s independence test for multivariate dependence [A simple measure of conditional dependence]," Biometrika, Biometrika Trust, vol. 109(3), pages 569-587.
    2. Segers, Johan, 2022. "Graphical and uniform consistency of estimated optimal transport plans," LIDAM Discussion Papers ISBA 2022022, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    3. Fang Han, 2024. "An Introduction to Permutation Processes (version 0.5)," Papers 2407.09664, arXiv.org.
    4. Hongjian Shi & Mathias Drton & Marc Hallin & Fang Han, 2023. "Semiparametrically Efficient Tests of Multivariate Independence Using Center-Outward Quadrant, Spearman, and Kendall Statistics," Working Papers ECARES 2023-03, ULB -- Universite Libre de Bruxelles.
    5. Marc Hallin & H Lui & Thomas Verdebout, 2022. "Nonparametric Measure-transportation-based Methods for Directional Data," Working Papers ECARES 2022-18, ULB -- Universite Libre de Bruxelles.

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