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Two-Sample Covariance Matrix Testing and Support Recovery in High-Dimensional and Sparse Settings

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  • Tony Cai
  • Weidong Liu
  • Yin Xia

Abstract

In the high-dimensional setting, this article considers three interrelated problems: (a) testing the equality of two covariance matrices and ; (b) recovering the support of ; and (c) testing the equality of and row by row. We propose a new test for testing the hypothesis H 0 : and investigate its theoretical and numerical properties. The limiting null distribution of the test statistic is derived and the power of the test is studied. The test is shown to enjoy certain optimality and to be especially powerful against sparse alternatives. The simulation results show that the test significantly outperforms the existing methods both in terms of size and power. Analysis of a prostate cancer dataset is carried out to demonstrate the application of the testing procedures. When the null hypothesis of equal covariance matrices is rejected, it is often of significant interest to further investigate how they differ from each other. Motivated by applications in genomics, we also consider recovering the support of and testing the equality of the two covariance matrices row by row. New procedures are introduced and their properties are studied. Applications to gene selection are also discussed. Supplementary materials for this article are available online.

Suggested Citation

  • 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.
    • Handle: RePEc:taf:jnlasa:v:108:y:2013:i:501:p:265-277
    DOI: 10.1080/01621459.2012.758041
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    References listed on IDEAS

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    1. Cai, Tony & Liu, Weidong, 2011. "Adaptive Thresholding for Sparse Covariance Matrix Estimation," Journal of the American Statistical Association, American Statistical Association, vol. 106(494), pages 672-684.
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