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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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    File URL: http://www.cirje.e.u-tokyo.ac.jp/research/dp/2014/2014cf933.pdf
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    References listed on IDEAS

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    1. Rauf Ahmad, M. & Werner, C. & Brunner, E., 2008. "Analysis of high-dimensional repeated measures designs: The one sample case," Computational Statistics & Data Analysis, Elsevier, vol. 53(2), pages 416-427, December.
    2. 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.
    3. Baik, Jinho & Silverstein, Jack W., 2006. "Eigenvalues of large sample covariance matrices of spiked population models," Journal of Multivariate Analysis, Elsevier, vol. 97(6), pages 1382-1408, July.
    4. Alexei Onatski & Marcelo Moreira J. & Marc Hallin, 2011. "Asymptotic Power of Sphericity Tests for High-Dimensional Data," Working Papers ECARES ECARES 2011-018, ULB -- Universite Libre de Bruxelles.
    5. Fan, Jianqing & Hall, Peter & Yao, Qiwei, 2007. "To How Many Simultaneous Hypothesis Tests Can Normal, Student's t or Bootstrap Calibration Be Applied?," Journal of the American Statistical Association, American Statistical Association, vol. 102, pages 1282-1288, December.
    6. Srivastava, Muni S., 2009. "A test for the mean vector with fewer observations than the dimension under non-normality," Journal of Multivariate Analysis, Elsevier, vol. 100(3), pages 518-532, March.
    7. Chen, Songxi, 2012. "Two Sample Tests for High Dimensional Covariance Matrices," MPRA Paper 46026, University Library of Munich, Germany.
    8. Schott, James R., 2007. "A test for the equality of covariance matrices when the dimension is large relative to the sample sizes," Computational Statistics & Data Analysis, Elsevier, vol. 51(12), pages 6535-6542, August.
    9. Srivastava, Muni S. & Kubokawa, Tatsuya, 2013. "Tests for multivariate analysis of variance in high dimension under non-normality," Journal of Multivariate Analysis, Elsevier, vol. 115(C), pages 204-216.
    10. Srivastava, Muni S. & Kollo, Tõnu & von Rosen, Dietrich, 2011. "Some tests for the covariance matrix with fewer observations than the dimension under non-normality," Journal of Multivariate Analysis, Elsevier, vol. 102(6), pages 1090-1103, July.
    11. 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.
    12. Srivastava, Muni S. & Du, Meng, 2008. "A test for the mean vector with fewer observations than the dimension," Journal of Multivariate Analysis, Elsevier, vol. 99(3), pages 386-402, March.
    13. Srivastava, Muni S. & Katayama, Shota & Kano, Yutaka, 2013. "A two sample test in high dimensional data," Journal of Multivariate Analysis, Elsevier, vol. 114(C), pages 349-358.
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    Cited by:

    1. 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.
    2. 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.
    3. Mao, Guangyu, 2016. "A note on tests for high-dimensional covariance matrices," Statistics & Probability Letters, Elsevier, vol. 117(C), pages 89-92.
    4. repec:taf:jnlasa:v:111:y:2016:i:514:p:721-735 is not listed on IDEAS
    5. Ana Angulo & Peter Burridge & Jesús Mur, 2017. "Testing for breaks in the weighting matrix," Documentos de Trabajo dt2017-01, Facultad de Ciencias Económicas y Empresariales, Universidad de Zaragoza.
    6. 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.
    7. 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.
    8. repec:eee:regeco:v:68:y:2018:i:c:p:115-129 is not listed on IDEAS

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