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A multivariate nonparametric test of independence

Author

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  • Bakirov, Nail K.
  • Rizzo, Maria L.
  • Szekely, Gábor J.

Abstract

A new nonparametric approach to the problem of testing the joint independence of two or more random vectors in arbitrary dimension is developed based on a measure of association determined by interpoint distances. The population independence coefficient takes values between 0 and 1, and equals zero if and only if the vectors are independent. We show that the corresponding statistic has a finite limit distribution if and only if the two random vectors are independent; thus we have a consistent test for independence. The coefficient is an increasing function of the absolute value of product moment correlation in the bivariate normal case, and coincides with the absolute value of correlation in the Bernoulli case. A simple modification of the statistic is affine invariant. The independence coefficient and the proposed statistic both have a natural extension to testing the independence of several random vectors. Empirical performance of the test is illustrated via a comparative Monte Carlo study.

Suggested Citation

  • Bakirov, Nail K. & Rizzo, Maria L. & Szekely, Gábor J., 2006. "A multivariate nonparametric test of independence," Journal of Multivariate Analysis, Elsevier, vol. 97(8), pages 1742-1756, September.
    • Handle: RePEc:eee:jmvana:v:97:y:2006:i:8:p:1742-1756
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    References listed on IDEAS

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    Cited by:

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    3. Jean-François Quessy, 2009. "Theoretical efficiency comparisons of independence tests based on multivariate versions of Spearman’s rho," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 70(3), pages 315-338, November.
    4. Marc Hallin, 2021. "Measure Transportation and Statistical Decision Theory," Working Papers ECARES 2021-04, ULB -- Universite Libre de Bruxelles.
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    7. Baringhaus, Ludwig & Gaigall, Daniel, 2015. "On an independence test approach to the goodness-of-fit problem," Journal of Multivariate Analysis, Elsevier, vol. 140(C), pages 193-208.
    8. Xiaojun Song & Haoyu Wei, 2021. "Nonparametric Tests of Conditional Independence for Time Series," Papers 2110.04847, arXiv.org.
    9. C Genest & J G Nešlehová & B Rémillard & O A Murphy, 2019. "Testing for independence in arbitrary distributions," Biometrika, Biometrika Trust, vol. 106(1), pages 47-68.
    10. Helmut Herwartz, 2022. "Modelling interaction patterns in a predator-prey system of two freshwater organisms in discrete time: an identified structural VAR approach," Statistical Methods & Applications, Springer;Società Italiana di Statistica, vol. 31(1), pages 63-85, March.
    11. Székely, Gábor J. & Rizzo, Maria L., 2013. "The distance correlation t-test of independence in high dimension," Journal of Multivariate Analysis, Elsevier, vol. 117(C), pages 193-213.
    12. Fan, Yanan & de Micheaux, Pierre Lafaye & Penev, Spiridon & Salopek, Donna, 2017. "Multivariate nonparametric test of independence," Journal of Multivariate Analysis, Elsevier, vol. 153(C), pages 189-210.
    13. Marie Hušková & Simos Meintanis, 2008. "Tests for the multivariate -sample problem based on the empirical characteristic function," Journal of Nonparametric Statistics, Taylor & Francis Journals, vol. 20(3), pages 263-277.
    14. Chu, Ba, 2023. "A distance-based test of independence between two multivariate time series," Journal of Multivariate Analysis, Elsevier, vol. 195(C).
    15. Györfi, László & Walk, Harro, 2012. "Strongly consistent nonparametric tests of conditional independence," Statistics & Probability Letters, Elsevier, vol. 82(6), pages 1145-1150.
    16. Matsushita, Raul & Figueiredo, Annibal & Da Silva, Sergio, 2012. "A suggested statistical test for measuring bivariate nonlinear dependence," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 391(20), pages 4891-4898.
    17. Kalemkerian, Juan & Fernández, Diego, 2020. "An independence test based on recurrence rates," Journal of Multivariate Analysis, Elsevier, vol. 178(C).

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