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Testing proportionality of two high-dimensional covariance matrices

Author

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  • Cheng, Guanghui
  • Liu, Baisen
  • Tian, Guoliang
  • Zheng, Shurong

Abstract

This article proposes three tests for proportionality hypotheses regrading high-dimensional covariance matrices. Compared with currently available tests in the literature that fail in situations involving a “large p small n” or require knowledge of the underlying normal distributions, these tests are nonparametric, and do not require specifying any known distribution to derive asymptotic distributions under both the null hypothesis as well as an alternative hypothesis. The theoretical justification for the proposed tests is provided to ensure their validity, especially when the number of dimensions p is larger than the sample size n. Numerical studies show that the proposed tests are adaptively powerful against dense as well as sparse alternatives for a wide range of dimensions and sample sizes. The tests were used to analyze a gene expression dataset to verify their effectiveness.

Suggested Citation

  • Cheng, Guanghui & Liu, Baisen & Tian, Guoliang & Zheng, Shurong, 2020. "Testing proportionality of two high-dimensional covariance matrices," Computational Statistics & Data Analysis, Elsevier, vol. 150(C).
    • Handle: RePEc:eee:csdana:v:150:y:2020:i:c:s0167947320300530
    DOI: 10.1016/j.csda.2020.106962
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    References listed on IDEAS

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    1. Tony Cai & Weidong Liu & Yin Xia, 2013. "Two-Sample Covariance Matrix Testing and Support Recovery in High-Dimensional and Sparse Settings," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 108(501), pages 265-277, March.
    2. Schott, James R., 1999. "A test for proportional covariance matrices," Computational Statistics & Data Analysis, Elsevier, vol. 32(2), pages 135-146, December.
    3. Flury, Bernhard K., 1986. "Proportionality of k covariance matrices," Statistics & Probability Letters, Elsevier, vol. 4(1), pages 29-33, January.
    4. 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.
    5. Liu, Baisen & Xu, Lin & Zheng, Shurong & Tian, Guo-Liang, 2014. "A new test for the proportionality of two large-dimensional covariance matrices," Journal of Multivariate Analysis, Elsevier, vol. 131(C), pages 293-308.
    6. Chen, Songxi, 2012. "Two Sample Tests for High Dimensional Covariance Matrices," MPRA Paper 46026, University Library of Munich, Germany.
    7. Xu, Lin & Liu, Baisen & Zheng, Shurong & Bao, Shaokun, 2014. "Testing proportionality of two large-dimensional covariance matrices," Computational Statistics & Data Analysis, Elsevier, vol. 78(C), pages 43-55.
    8. Jinyuan Chang & Wen Zhou & Wen-Xin Zhou & Lan Wang, 2017. "Comparing large covariance matrices under weak conditions on the dependence structure and its application to gene clustering," Biometrics, The International Biometric Society, vol. 73(1), pages 31-41, March.
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    1. Ahmad, Rauf, 2022. "Tests for proportionality of matrices with large dimension," Journal of Multivariate Analysis, Elsevier, vol. 189(C).

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