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A distance covariance test of independence in high dimension, low sample size contexts

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  • Kai Xu

    (Anhui Normal University)

  • Minghui Yang

    (Anhui Normal University)

Abstract

To check the mutual independence of a high-dimensional random vector without Gaussian assumption, Yao et al. (Journal of the Royal Statistical Society Series B, 80,455–480, 2018) recently introduced an important test by virtue of pairwise distance covariances. Despite its usefulness, the state-of-art test tends to have unsatisfactory size performance when the sample size is small. The present paper provides a theoretical explanation about this phenomenon, and accordingly proposes a new test in high dimension, low sample size contexts. The new test can be even justified as the dimension tends to infinity, regardless of whether the sample size is fixed or diverges. The power of the proposed distance covariance test is also investigated. To examine our theoretical findings and check the performance of the new test, simulation studies are applied. We further illustrate the proposed method by empirical analysis of a real dataset.

Suggested Citation

  • Kai Xu & Minghui Yang, 2025. "A distance covariance test of independence in high dimension, low sample size contexts," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 77(5), pages 731-755, October.
    • Handle: RePEc:spr:aistmt:v:77:y:2025:i:5:d:10.1007_s10463-025-00928-x
    DOI: 10.1007/s10463-025-00928-x
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    References listed on IDEAS

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    1. Peter Hall & J. S. Marron & Amnon Neeman, 2005. "Geometric representation of high dimension, low sample size data," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 67(3), pages 427-444, June.
    2. Guangyu Mao, 2020. "On high-dimensional tests for mutual independence based on Pearson’s correlation coefficient," Communications in Statistics - Theory and Methods, Taylor & Francis Journals, vol. 49(14), pages 3572-3584, July.
    3. 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.
    4. Runze Li & Wei Zhong & Liping Zhu, 2012. "Feature Screening via Distance Correlation Learning," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 107(499), pages 1129-1139, September.
    5. James R. Schott, 2005. "Testing for complete independence in high dimensions," Biometrika, Biometrika Trust, vol. 92(4), pages 951-956, December.
    6. Niklas Pfister & Peter Bühlmann & Bernhard Schölkopf & Jonas Peters, 2018. "Kernel‐based tests for joint independence," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 80(1), pages 5-31, January.
    7. 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.
    8. Chen, Song Xi & Zhang, Li-Xin & Zhong, Ping-Shou, 2010. "Tests for High-Dimensional Covariance Matrices," Journal of the American Statistical Association, American Statistical Association, vol. 105(490), pages 810-819.
    9. Xiaofeng Shao & Jingsi Zhang, 2014. "Martingale Difference Correlation and Its Use in High-Dimensional Variable Screening," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 109(507), pages 1302-1318, September.
    10. Shun Yao & Xianyang Zhang & Xiaofeng Shao, 2018. "Testing mutual independence in high dimension via distance covariance," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 80(3), pages 455-480, June.
    11. Jeongyoun Ahn & J. S. Marron & Keith M. Muller & Yueh-Yun Chi, 2007. "The high-dimension, low-sample-size geometric representation holds under mild conditions," Biometrika, Biometrika Trust, vol. 94(3), pages 760-766.
    12. Shubhadeep Chakraborty & Xianyang Zhang, 2019. "Distance Metrics for Measuring Joint Dependence with Application to Causal Inference," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 114(528), pages 1638-1650, October.
    13. Xu, Kai & Cheng, Qing & He, Daojiang, 2025. "On summed nonparametric dependence measures in high dimensions, fixed or large samples," Computational Statistics & Data Analysis, Elsevier, vol. 205(C).
    14. Aki Ishii & Kazuyoshi Yata & Makoto Aoshima, 2021. "Hypothesis tests for high-dimensional covariance structures," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 73(3), pages 599-622, June.
    15. Fang Han & Shizhe Chen & Han Liu, 2017. "Distribution-free tests of independence in high dimensions," Biometrika, Biometrika Trust, vol. 104(4), pages 813-828.
    16. Mao, Guangyu, 2018. "Testing independence in high dimensions using Kendall’s tau," Computational Statistics & Data Analysis, Elsevier, vol. 117(C), pages 128-137.
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