Corrections to LRT on large-dimensional covariance matrix by RMT
Zhidong Bai, Dandan Jiang, Jian-Feng Yao, and Shurong Zheng
Source: Ann. Statist.
Volume 37, Number 6B
(2009), 3822-3840.
Abstract
In this paper, we give an explanation to the failure of two likelihood ratio procedures for testing about covariance matrices from Gaussian populations when the dimension p is large compared to the sample size n. Next, using recent central limit theorems for linear spectral statistics of sample covariance matrices and of random F-matrices, we propose necessary corrections for these LR tests to cope with high-dimensional effects. The asymptotic distributions of these corrected tests under the null are given. Simulations demonstrate that the corrected LR tests yield a realized size close to nominal level for both moderate p (around 20) and high dimension, while the traditional LR tests with χ2 approximation fails.
Another contribution from the paper is that for testing the equality between two covariance matrices, the proposed correction applies equally for non-Gaussian populations yielding a valid pseudo-likelihood ratio test.
Primary Subjects: 62H15
Secondary Subjects: 62H10
Keywords: High-dimensional data; testing on covariance matrices; Marčenko–Pastur distributions; random F-matrices
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Links and Identifiers
Permanent link to this document: http://projecteuclid.org/euclid.aos/1256303528
Digital Object Identifier: doi:10.1214/09-AOS694
References
[1] Bai, Z. D. and Saranadasa, H. (1996). Effect of high dimension comparison of significance tests for a high-dimensional two sample problem. Statist. Sinica 6 311–329.
[2] Bai, Z. D., Jiang, D., Yao, J.-F. and Zheng, S. (2008). Corrections to LRT on large dimensional covariance matrix by RMT (full version). Preprint. Available at arXiv:0902.0552.
[3] Bai, Z. D. and Silverstein, J. W. (2004). CLT for linear spectral statistics of large-dimensional sample covariance matrices. Ann. Probab. 32 553–605.
[4] Bai, Z. D. and Silverstein, J. W. (2006). Spectral Analysis of Large-Dimensional Random Matrices, 1st ed. Science Press, Beijing.
[5] Dempster, A. P. (1958). A high-dimensional two sample significance test. Ann. Math. Statist. 29 995–1010.
Mathematical Reviews (MathSciNet):
MR112207
[6] Ledoit, O. and Wolf, M. (2002). Some hypothesis tests for the covariance matrix when the dimension is large compared to the sample size. Ann. Statist. 30 1081–1102.
[7] Pastur, L. and Lytova, A. (2008). Central limit theorem for linear eigenvalue statistics of random matrices with independent entries. Preprint. Available at arXiv:0809.4698v1.
[8] Schott, J. R. (2007). A test for the equality of covariance matrices when the dimension is large relative to the sample size. Comput. Statist. Data Anal. 51 6535–6542.
[9] Srivastava, M. S. (2005). Some tests concerning the covariance matrix in high-dimensional data. J. Japan Statist. Soc. 35 251–272.
[10] Zheng, S. (2008). Central limit theorem for linear spectral statistics of large dimensional F-matrix. Preprint. Northeast Normal Univ., Changchun, China.
