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Empirical likelihood test for the equality of several high-dimensional covariance matrices

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Abstract

The testing covariance equality is of importance in many areas of statistical analysis, such as microarray analysis and quality control. Conventional tests for the finite-dimensional covariance do not apply to high-dimensional data in general, and tests for the high-dimensional covariance in the literature usually depend on some special structure of the matrix and whether the dimension diverges. In this paper, we propose a jack-knife empirical likelihood method to test the equality of covariance matrices. The asymptotic distribution of the new test is regardless of the divergent or fixed dimension. Simulation studies show that the new test has a very stable size with respect to the dimension and it is also more powerful than the test proposed by Schott (2007) and studied by Srivastava and Yanagihara (2010). Furthermore, we illustrate the method using a breast cancer dataset.

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References

  1. Bai Z D, Jiang D, Yao J F, et al. Corrections to LRT on large dimensional covariance matrix by RMT. Ann Statist, 2009, 37: 3822–3840

    Article  MathSciNet  Google Scholar 

  2. Bai Z D, Saranadasa H. Effect of high dimension: By an example of a two sample problem. Statist Sinica, 1996, 6: 311–329

    MathSciNet  MATH  Google Scholar 

  3. Bickel P, Levina E. Regularized estimation of large covariance matrices. Ann Statist, 2008, 36: 199–227

    MathSciNet  MATH  Google Scholar 

  4. Bickel P, Levina E. Covariance regularization by thresholding. Ann Statist, 2008, 36: 2577–2604

    MathSciNet  MATH  Google Scholar 

  5. Cai T, Liu W D. Adaptive thresholding for sparse covariance matrix estimation. J Amer Statist Assoc, 2011, 106: 672–684

    Article  MathSciNet  Google Scholar 

  6. Cai T T, Jiang T F. Limiting laws of coherence of random matrices with applications to testing covariance structure and construction of compressed sensing matrices. Ann Statist, 2011, 39: 1496–1525

    MathSciNet  MATH  Google Scholar 

  7. Cai T T, Ma Z. Optimal hypothesis testing for high dimensional covariance matrices. Bernoulli, 2013, 19: 2359–2388

    Article  MathSciNet  Google Scholar 

  8. Cai T T, Zhang C H, Zhou H. Optimal rates of convergence for covariance matrix estimation. Ann Statist, 2010, 38: 2118–2144

    MathSciNet  MATH  Google Scholar 

  9. Chen S X, Qin Y L. A two-sample test for high-dimensional data with applications to gene-set testing. Ann Statist, 2010, 38: 808–835

    MathSciNet  MATH  Google Scholar 

  10. Chen S X, Zhang L X, Zhong P S. Tests for high-dimensional covariance matrices. J Amer Statist Assoc, 2010, 105: 810–819

    Article  MathSciNet  Google Scholar 

  11. Dharmadhikari S W, Fabian V, Jogdeo K. Bounds on the moments of martingale. Ann Math Statist, 1968, 39: 1719–1723

    Article  MathSciNet  Google Scholar 

  12. Fan J, Fan Y, Lv J. High dimensional covariance matrix estimation using a factor model. J Econometrics, 2008, 147: 186–197

    Article  MathSciNet  Google Scholar 

  13. Jiang T F, Yang F. Central limit theorems for classical likelihood ratio tests for high-dimensional normal distributions. Ann Statist, 2013, 41: 2029–2074

    MathSciNet  MATH  Google Scholar 

  14. Ledoit O, Wolf M. Some hypothesis tests for the covariance matrix when the dimension is large compared to the sample size. Ann Statist, 2002, 30: 1081–1102

    Article  MathSciNet  Google Scholar 

  15. Muirhead R J. On the distribution of the likelihood ratio test of equality of normal populations. Canad J Statist, 1982, 10: 59–62

    Article  MathSciNet  Google Scholar 

  16. Owen A. Empirical likelihood ratio confidence regions. Ann Statist, 1990, 18: 90–120

    Article  MathSciNet  Google Scholar 

  17. Qin J, Lawless J. Empirical likelihood and tests of parameters subject to constraints. Canad J Statist, 1995, 23: 145–149

    Article  MathSciNet  Google Scholar 

  18. Rothman A J, Levina E, Zhu J. Generalized thresholding of large covariance matrices. J Amer Statist Assoc, 2009, 104: 177–186

    Article  MathSciNet  Google Scholar 

  19. Schott J R. Some tests for the equality of covariance matrices. J Statist Plann Inference, 2001, 94: 25–36

    Article  MathSciNet  Google Scholar 

  20. Schott J R. A test for the equality of covariance matrices when the dimension is large relative to the sample sizes. Comput Statist Data Anal, 2007, 51: 6535–6542

    Article  MathSciNet  Google Scholar 

  21. Srivastava M S, Yanagihara H. Testing the equality of several covariance matrices with fewer observations than the dimension. J Multivariate Anal, 2010, 101: 1319–1329

    Article  MathSciNet  Google Scholar 

  22. von Bahr B, Esseen C G. Inequality for the rth absolute moment of a sum of random variables, 1 ⩽ r ⩽ 2. Ann Math Statist, 1965, 36: 299–393

    Article  MathSciNet  Google Scholar 

  23. Zhang R M, Peng L, Wang R D. Tests for covariance matrix with fixed or divergent dimension. Ann Statist, 2013, 41: 2075–2096

    MathSciNet  MATH  Google Scholar 

  24. Zheng S R, Lin R T, Guo J H, et al. Testing homogeneity of high-dimensional covariance matrices. Statist Sinica, 2020, 30: 35–53

    MathSciNet  MATH  Google Scholar 

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Acknowledgements

This work was supported by the Simons Foundation, National Natural Science Foundation of China (Grant Nos. 11771390 and 11371318), Zhejiang Provincial Natural Science Foundation of China (Grant No. LR16A010001), the University of Sydney and Zhejiang University Partnership Collaboration Awards and the Fundamental Research Funds for the Central Universities. The authors thank the associate editor and two anonymous referees for constructive comments and helpful suggestions, which led to a substantial improvement of this paper.

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Authors and Affiliations

  1. School of Mathematical Science, Zhejiang University, Hangzhou, 310027, China

    Guili Liao & Rongmao Zhang

  2. Department of Risk Management and Insurance, Georgia State University, Atlanta, GA, 30303, USA

    Liang Peng

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  1. Guili Liao
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  2. Liang Peng
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  3. Rongmao Zhang
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Correspondence to Rongmao Zhang.

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Liao, G., Peng, L. & Zhang, R. Empirical likelihood test for the equality of several high-dimensional covariance matrices. Sci. China Math. 64, 2775–2792 (2021). https://doi.org/10.1007/s11425-019-1688-1

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  • DOI: https://doi.org/10.1007/s11425-019-1688-1

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