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Distribution-free tests of independence in high dimensions

Author

Listed:
  • Fang Han
  • Shizhe Chen
  • Han Liu

Abstract

SummaryWe consider the testing of mutual independence among all entries in a $d$-dimensional random vector based on $n$ independent observations. We study two families of distribution-free test statistics, which include Kendall’s tau and Spearman’s rho as important examples. We show that under the null hypothesis the test statistics of these two families converge weakly to Gumbel distributions, and we propose tests that control the Type I error in the high-dimensional setting where $d >n$. We further show that the two tests are rate-optimal in terms of power against sparse alternatives and that they outperform competitors in simulations, especially when $d$ is large.

Suggested Citation

  • Fang Han & Shizhe Chen & Han Liu, 2017. "Distribution-free tests of independence in high dimensions," Biometrika, Biometrika Trust, vol. 104(4), pages 813-828.
    • Handle: RePEc:oup:biomet:v:104:y:2017:i:4:p:813-828.
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    File URL: http://hdl.handle.net/10.1093/biomet/asx050
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    Citations

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    Cited by:

    1. Wu, Zeyu & Wang, Cheng, 2022. "Limiting spectral distribution of large dimensional Spearman’s rank correlation matrices," Journal of Multivariate Analysis, Elsevier, vol. 191(C).
    2. Karch, Julian D. & Perez-Alonso, Andres F. & Bergsma, Wicher P., 2024. "Beyond Pearson’s correlation: modern nonparametric independence tests for psychological research," LSE Research Online Documents on Economics 124587, London School of Economics and Political Science, LSE Library.
    3. Ivair R. Silva & Yan Zhuang & Julio C. A. da Silva Junior, 2022. "Kronecker delta method for testing independence between two vectors in high-dimension," Statistical Papers, Springer, vol. 63(2), pages 343-365, April.
    4. Wang, Hongfei & Liu, Binghui & Feng, Long & Ma, Yanyuan, 2024. "Rank-based max-sum tests for mutual independence of high-dimensional random vectors," Journal of Econometrics, Elsevier, vol. 238(1).
    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.
    6. Jin, Ze & Matteson, David S., 2018. "Generalizing distance covariance to measure and test multivariate mutual dependence via complete and incomplete V-statistics," Journal of Multivariate Analysis, Elsevier, vol. 168(C), pages 304-322.
    7. Dörnemann, Nina, 2023. "Likelihood ratio tests under model misspecification in high dimensions," Journal of Multivariate Analysis, Elsevier, vol. 193(C).

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