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Limit theorem associated with Wishart matrices with application to hypothesis testing for common principal components

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  • Tsukuda, Koji
  • Matsuura, Shun

Abstract

This paper describes the derivation of a new property of the Wishart distribution when the degrees of freedom and the sizes of scale matrices grow simultaneously. In particular, the asymptotic normality of the trace of the product of four independent Wishart matrices is demonstrated for a high-dimensional asymptotic regime. As an application of the result, a statistical test procedure for the common principal components hypothesis is proposed. For this problem, the proposed test statistic is asymptotically normal under the null hypothesis and diverges to positive infinity in probability under the alternative hypothesis.

Suggested Citation

  • 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).
    • Handle: RePEc:eee:jmvana:v:186:y:2021:i:c:s0047259x21001007
    DOI: 10.1016/j.jmva.2021.104822
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    References listed on IDEAS

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    1. 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.
    2. 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.
    3. 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.
    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. 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).
    6. Tsukuda, Koji & Matsuura, Shun, 2019. "High-dimensional testing for proportional covariance matrices," Journal of Multivariate Analysis, Elsevier, vol. 171(C), pages 412-420.
    7. 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.
    8. Marc Hallin & Davy Paindaveine & Thomas Verdebout, 2010. "Testing for Common Principal Components under Heterokurticity," Journal of Nonparametric Statistics, Taylor & Francis Journals, vol. 22(7), pages 879-895.
    9. Chen, Songxi, 2012. "Two Sample Tests for High Dimensional Covariance Matrices," MPRA Paper 46026, University Library of Munich, Germany.
    10. 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.
    11. Srivastava, Muni S. & Yanagihara, Hirokazu, 2010. "Testing the equality of several covariance matrices with fewer observations than the dimension," Journal of Multivariate Analysis, Elsevier, vol. 101(6), pages 1319-1329, July.
    12. Robert J. Boik, 2002. "Spectral models for covariance matrices," Biometrika, Biometrika Trust, vol. 89(1), pages 159-182, March.
    13. 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.
    14. 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.
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