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Testing for independence of high-dimensional variables: ρV-coefficient based approach

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  • Hyodo, Masashi
  • Nishiyama, Takahiro
  • Pavlenko, Tatjana

Abstract

We treat the problem of testing mutual independence of k high-dimensional random vectors when the data are multivariate normal and k≥2 is a fixed integer. For this purpose, we focus on the vector correlation coefficient, ρV and propose an extension of its classical estimator which is constructed to correct potential sources of inconsistency related to the high dimensionality. Building on the proposed estimator of ρV, we derive the new test statistic and study its limiting behavior in a general high-dimensional asymptotic framework which allows the vector’s dimensionality arbitrarily exceed the sample size. Specifically, we show that the asymptotic distribution of the test statistic under the main hypothesis of independence is standard normal and that the proposed test is size and power consistent. Using our statistics, we further construct the step-down multiple comparison procedure based on the closed testing strategy for the simultaneous test for independence. Accuracy of the proposed tests in finite samples is shown through simulations for a variety of high-dimensional scenarios in combination with a number of alternative dependence structures. Real data analysis is performed to illustrate the utility of the test procedures.

Suggested Citation

  • Hyodo, Masashi & Nishiyama, Takahiro & Pavlenko, Tatjana, 2020. "Testing for independence of high-dimensional variables: ρV-coefficient based approach," Journal of Multivariate Analysis, Elsevier, vol. 178(C).
    • Handle: RePEc:eee:jmvana:v:178:y:2020:i:c:s0047259x20302086
    DOI: 10.1016/j.jmva.2020.104627
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    References listed on IDEAS

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    1. Srivastava, Muni S. & Reid, N., 2012. "Testing the structure of the covariance matrix with fewer observations than the dimension," Journal of Multivariate Analysis, Elsevier, vol. 112(C), pages 156-171.
    2. 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.
    3. G. Pan & J. Gao & Y. Yang & M. Guo, 2012. "Independence Test for High Dimensional Random Vectors," Monash Econometrics and Business Statistics Working Papers 1/12, Monash University, Department of Econometrics and Business Statistics.
    4. Harrar, Solomon W. & Kong, Xiaoli, 2016. "High-dimensional multivariate repeated measures analysis with unequal covariance matrices," Journal of Multivariate Analysis, Elsevier, vol. 145(C), pages 1-21.
    5. Masashi Hyodo & Nobumichi Shutoh & Takahiro Nishiyama & Tatjana Pavlenko, 2015. "Testing block-diagonal covariance structure for high-dimensional data," Statistica Neerlandica, Netherlands Society for Statistics and Operations Research, vol. 69(4), pages 460-482, November.
    6. Josse, J. & Pagès, J. & Husson, F., 2008. "Testing the significance of the RV coefficient," Computational Statistics & Data Analysis, Elsevier, vol. 53(1), pages 82-91, September.
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    Cited by:

    1. Tsukuda, Koji & Matsuura, Shun, 2021. "Limit theorem associated with Wishart matrices with application to hypothesis testing for common principal components," Journal of Multivariate Analysis, Elsevier, vol. 186(C).
    2. Qiu, Tao & Xu, Wangli & Zhu, Lixing, 2023. "Independence tests with random subspace of two random vectors in high dimension," Journal of Multivariate Analysis, Elsevier, vol. 195(C).

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