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Symmetric rank covariances: a generalized framework for nonparametric measures of dependence

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  • L Weihs
  • M Drton
  • N Meinshausen

Abstract

SummaryThe need to test whether two random vectors are independent has spawned many competing measures of dependence. We focus on nonparametric measures that are invariant under strictly increasing transformations, such as Kendall’s tau, Hoeffding’s $D$, and the Bergsma–Dassios sign covariance. Each exhibits symmetries that are not readily apparent from their definitions. Making these symmetries explicit, we define a new class of multivariate nonparametric measures of dependence that we call symmetric rank covariances. This new class generalizes the above measures and leads naturally to multivariate extensions of the Bergsma–Dassios sign covariance. Symmetric rank covariances may be estimated unbiasedly using U-statistics, for which we prove results on computational efficiency and large-sample behaviour. The algorithms we develop for their computation include, to the best of our knowledge, the first efficient algorithms for Hoeffding’s $D$ statistic in the multivariate setting.

Suggested Citation

  • 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.
  • Handle: RePEc:oup:biomet:v:105:y:2018:i:3:p:547-562.
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    File URL: http://hdl.handle.net/10.1093/biomet/asy021
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    References listed on IDEAS

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    1. Luca Weihs & Mathias Drton & Dennis Leung, 2016. "Efficient computation of the Bergsma–Dassios sign covariance," Computational Statistics, Springer, vol. 31(1), pages 315-328, March.
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    3. Hengjian Cui & Runze Li & Wei Zhong, 2015. "Model-Free Feature Screening for Ultrahigh Dimensional Discriminant Analysis," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 110(510), pages 630-641, June.
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    5. Liping Zhu & Kai Xu & Runze Li & Wei Zhong, 2017. "Projection correlation between two random vectors," Biometrika, Biometrika Trust, vol. 104(4), pages 829-843.
    6. Ruth Heller & Yair Heller & Malka Gorfine, 2013. "A consistent multivariate test of association based on ranks of distances," Biometrika, Biometrika Trust, vol. 100(2), pages 503-510.
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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. H Shi & M Drton & F Han, 2022. "On the power of Chatterjee’s rank correlation [Adaptive test of independence based on HSIC measures]," Biometrika, Biometrika Trust, vol. 109(2), pages 317-333.
    3. Wu, Zeyu & Wang, Cheng, 2022. "Limiting spectral distribution of large dimensional Spearman’s rank correlation matrices," Journal of Multivariate Analysis, Elsevier, vol. 191(C).
    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. Hongjian Shi & Marc Hallin & Mathias Drton & Fang Han, 2020. "Rate-Optimality of Consistent Distribution-Free Tests of Independence Based on Center-Outward Ranks and Signs," Working Papers ECARES 2020-23, ULB -- Universite Libre de Bruxelles.

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