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

Author

Listed:
  • Hongjian Shi
  • Mathias Drton
  • Marc Hallin
  • Fang Han

Abstract

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 uniformreference 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 theclassical Gaussian procedures.

Suggested Citation

  • 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.
  • Handle: RePEc:eca:wpaper:2013/355918
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    References listed on IDEAS

    as
    1. Haeun Moon & Kehui Chen, 2022. "Interpoint-ranking sign covariance for the test of independence [Prediction by supervised principal components]," Biometrika, Biometrika Trust, vol. 109(1), pages 165-179.
    2. Victor Chernozhukov & Alfred Galichon & Marc Hallin & Marc Henry, 2014. "Monge-Kantorovich Depth, Quantiles, Ranks, and Signs," Papers 1412.8434, arXiv.org, revised Sep 2015.
    3. M. Hallin & D. La Vecchia & H. Liu, 2022. "Center-Outward R-Estimation for Semiparametric VARMA Models," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 117(538), pages 925-938, April.
    4. Taskinen, Sara & Kankainen, Annaliisa & Oja, Hannu, 2003. "Sign test of independence between two random vectors," Statistics & Probability Letters, Elsevier, vol. 62(1), pages 9-21, March.
    5. Guillaume Carlier & Victor Chernozhukov & Alfred Galichon, 2015. "Vector quantile regression: an optimal transport approach," CeMMAP working papers 58/15, Institute for Fiscal Studies.
    6. Eustasio del Barrio & Alberto González-Sanz & Marc Hallin, 2022. "Nonparametric Multiple-Output Center-Outward Quantile Regression," Working Papers ECARES 2022-10, ULB -- Universite Libre de Bruxelles.
    7. Bernard Garel & Marc Hallin, 1995. "Local asymptotic normality of multivariate ARMA processes with a linear trend," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 47(3), pages 551-579, September.
    8. repec:hal:spmain:info:hdl:2441/64itsev5509q8aa5mrbhi0g0b6 is not listed on IDEAS
    9. repec:hal:spmain:info:hdl:2441/4c5431jp6o888pdrcs0fuirl40 is not listed on IDEAS
    10. 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.
    11. Guillaume Carlier & Victor Chernozhukov & Alfred Galichon, 2016. "Vector Quantile Regression: An Optimal Transport Approach," SciencePo Working papers hal-03567920, HAL.
    12. Taskinen, Sara & Oja, Hannu & Randles, Ronald H., 2005. "Multivariate Nonparametric Tests of Independence," Journal of the American Statistical Association, American Statistical Association, vol. 100, pages 916-925, September.
    13. Marc Hallin, 2017. "On Distribution and Quantile Functions, Ranks and Signs in R_d," Working Papers ECARES ECARES 2017-34, ULB -- Universite Libre de Bruxelles.
    14. L Weihs & M Drton & N Meinshausen, 2018. "Symmetric rank covariances: a generalized framework for nonparametric measures of dependence," Biometrika, Biometrika Trust, vol. 105(3), pages 547-562.
    15. Eustasio Del Barrio & Juan Cuesta Albertos & Marc Hallin & Carlos Matran, 2018. "Smooth Cyclically Monotone Interpolation and Empirical Center-Outward Distribution Functions," Working Papers ECARES 2018-15, ULB -- Universite Libre de Bruxelles.
    16. Marc Hallin & Gilles Mordant, 2021. "On the Finite-Sample Performance of Measure Transportation-Based Multivariate Rank Tests," Working Papers ECARES 2021-24, ULB -- Universite Libre de Bruxelles.
    17. Victor Chernozhukov & Alfred Galichon & Marc Hallin & Marc Henry, 2014. "Monge-Kantorovich Depth, Quantiles, Ranks, and Signs," Papers 1412.8434, arXiv.org, revised Sep 2015.
    18. Carlier, Guillaume & Chernozhukov, Victor & Galichon, Alfred, 2017. "Vector quantile regression beyond the specified case," Journal of Multivariate Analysis, Elsevier, vol. 161(C), pages 96-102.
    Full references (including those not matched with items on IDEAS)

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    Keywords

    Semiparametrically Efficient Tests; Multivariate Independence; Center-Outward Quadrant; Spearman; Kendall Statistics;
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