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Tests for Covariance Matrices in High Dimension with Less Sample Size

Author

Listed:
  • Muni S. Srivastava

    (Department of Statistics, University of Toronto)

  • Hirokazu Yanagihara

    (Department of Mathematics, Hiroshima University)

  • Tatsuya Kubokawa

    (Faculty of Economics, The University of Tokyo)

Abstract

In this article, we propose tests for covariance matrices of high dimension with fewer observations than the dimension for a general class of distributions with positive definite covariance matrices. In one-sample case, tests are proposed for sphericity and for testing the hypothesis that the covariance matrix ∑ is an identity matrix, by providing an unbiased estimator of tr [∑ 2 ] under the general model which requires no more computing time than the one available in the literature for normal model. In the two-sample case, tests for the equality of two covariance matrices are given. The asymptotic distributions of proposed tests in one-sample case are derived under the assumption that the sample size N = O ( p δ ), 1/2 < δ < 1, where p is the dimension of the random vector, and O ( p δ ) means that N/p goes to zero as N and p go to infinity. Similar assumptions are made in the two-sample case.

Suggested Citation

  • Muni S. Srivastava & Hirokazu Yanagihara & Tatsuya Kubokawa, 2014. "Tests for Covariance Matrices in High Dimension with Less Sample Size," CIRJE F-Series CIRJE-F-933, CIRJE, Faculty of Economics, University of Tokyo.
    • Handle: RePEc:tky:fseres:2014cf933
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    References listed on IDEAS

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    Cited by:

    1. Wang, Zhendong & Xu, Xingzhong, 2021. "Testing high dimensional covariance matrices via posterior Bayes factor," Journal of Multivariate Analysis, Elsevier, vol. 181(C).
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    3. Butucea, Cristina & Zgheib, Rania, 2016. "Sharp minimax tests for large Toeplitz covariance matrices with repeated observations," Journal of Multivariate Analysis, Elsevier, vol. 146(C), pages 164-176.
    4. Li, Yang & Wang, Zhaojun & Zou, Changliang, 2016. "A simpler spatial-sign-based two-sample test for high-dimensional data," Journal of Multivariate Analysis, Elsevier, vol. 149(C), pages 192-198.
    5. Mao, Guangyu, 2016. "A note on tests for high-dimensional covariance matrices," Statistics & Probability Letters, Elsevier, vol. 117(C), pages 89-92.
    6. Ikeda, Yuki & Kubokawa, Tatsuya & Srivastava, Muni S., 2016. "Comparison of linear shrinkage estimators of a large covariance matrix in normal and non-normal distributions," Computational Statistics & Data Analysis, Elsevier, vol. 95(C), pages 95-108.
    7. Zheng, Bang Quan, 2021. "RGLS and RLS in Covariance Structure Analysis," SocArXiv aejgf, Center for Open Science.
    8. Long Feng & Changliang Zou & Zhaojun Wang, 2016. "Multivariate-Sign-Based High-Dimensional Tests for the Two-Sample Location Problem," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 111(514), pages 721-735, April.
    9. Tsukuda, Koji & Matsuura, Shun, 2019. "High-dimensional testing for proportional covariance matrices," Journal of Multivariate Analysis, Elsevier, vol. 171(C), pages 412-420.
    10. Yuki Ikeda & Tatsuya Kubokawa & Muni S. Srivastava, 2015. "Comparison of Linear Shrinkage Estimators of a Large Covariance Matrix in Normal and Non-normal Distributions," CIRJE F-Series CIRJE-F-970, CIRJE, Faculty of Economics, University of Tokyo.
    11. Angulo, Ana & Burridge, Peter & Mur, Jesús, 2018. "Testing for breaks in the weighting matrix," Regional Science and Urban Economics, Elsevier, vol. 68(C), pages 115-129.
    12. Yamada, Yuki & Hyodo, Masashi & Nishiyama, Takahiro, 2017. "Testing block-diagonal covariance structure for high-dimensional data under non-normality," Journal of Multivariate Analysis, Elsevier, vol. 155(C), pages 305-316.
    13. Xu, Kai & Hao, Xinxin, 2019. "A nonparametric test for block-diagonal covariance structure in high dimension and small samples," Journal of Multivariate Analysis, Elsevier, vol. 173(C), pages 551-567.
    14. Imbs, Jean & Pauwels, Laurent, 2019. "Fundamental Moments," Working Papers BAWP-2019-06, University of Sydney Business School, Discipline of Business Analytics.
    15. Peng, Liuhua & Chen, Song Xi & Zhou, Wen, 2016. "More powerful tests for sparse high-dimensional covariances matrices," Journal of Multivariate Analysis, Elsevier, vol. 149(C), pages 124-143.
    16. 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).
    17. Hayakawa, Kazuhiko, 2024. "Recent development of covariance structure analysis in economics," Econometrics and Statistics, Elsevier, vol. 29(C), pages 31-48.
    18. Deepak Nag Ayyala & Santu Ghosh & Daniel F. Linder, 2022. "Covariance matrix testing in high dimension using random projections," Computational Statistics, Springer, vol. 37(3), pages 1111-1141, July.
    19. Zhendong Wang & Xingzhong Xu, 2021. "High-dimensional sphericity test by extended likelihood ratio," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 84(8), pages 1169-1212, November.
    20. Laíla Luana Campos & Daniel Furtado Ferreira, 2022. "Robust modified classical spherical tests in the presence of outliers," Statistical Papers, Springer, vol. 63(5), pages 1561-1576, October.
    21. Masashi Hyodo & Takahiro Nishiyama, 2018. "A simultaneous testing of the mean vector and the covariance matrix among two populations for high-dimensional data," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 27(3), pages 680-699, September.

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