Bernoulli

  • Bernoulli
  • Volume 23, Number 2 (2017), 1130-1178.

CLT for eigenvalue statistics of large-dimensional general Fisher matrices with applications

Shurong Zheng, Zhidong Bai, and Jianfeng Yao

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Abstract

Random Fisher matrices arise naturally in multivariate statistical analysis and understanding the properties of its eigenvalues is of primary importance for many hypothesis testing problems like testing the equality between two covariance matrices, or testing the independence between sub-groups of a multivariate random vector. Most of the existing work on random Fisher matrices deals with a particular situation where the population covariance matrices are equal. In this paper, we consider general Fisher matrices with arbitrary population covariance matrices and develop their spectral properties when the dimensions are proportionally large compared to the sample size. The paper has two main contributions: first the limiting distribution of the eigenvalues of a general Fisher matrix is found and second, a central limit theorem is established for a wide class of functionals of these eigenvalues. Applications of the main results are also developed for testing hypotheses on high-dimensional covariance matrices.

Article information

Source
Bernoulli, Volume 23, Number 2 (2017), 1130-1178.

Dates
Received: September 2014
Revised: April 2015
First available in Project Euclid: 4 February 2017

Permanent link to this document
https://projecteuclid.org/euclid.bj/1486177395

Digital Object Identifier
doi:10.3150/15-BEJ772

Mathematical Reviews number (MathSciNet)
MR3606762

Zentralblatt MATH identifier
1375.60066

Keywords
central limit theorem equality of covariance matrices large-dimensional covariance matrices large-dimensional Fisher matrix linear spectral statistics

Citation

Zheng, Shurong; Bai, Zhidong; Yao, Jianfeng. CLT for eigenvalue statistics of large-dimensional general Fisher matrices with applications. Bernoulli 23 (2017), no. 2, 1130--1178. doi:10.3150/15-BEJ772. https://projecteuclid.org/euclid.bj/1486177395


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References

  • [1] Anderson, T.W. (2003). An Introduction to Multivariate Statistical Analysis, 3rd ed. Wiley Series in Probability and Statistics. Hoboken, NJ: Wiley.
  • [2] Bai, Z., Jiang, D., Yao, J.-F. and Zheng, S. (2009). Corrections to LRT on large-dimensional covariance matrix by RMT. Ann. Statist. 37 3822–3840.
  • [3] Bai, Z. and Saranadasa, H. (1996). Effect of high dimension: By an example of a two sample problem. Statist. Sinica 6 311–329.
  • [4] Bai, Z. and Silverstein, J.W. (2010). Spectral Analysis of Large Dimensional Random Matrices, 2nd ed. Springer Series in Statistics. New York: Springer.
  • [5] Bai, Z.D. (1984). A note on asymptotic joint distribution of the eigenvalues of a noncentral multivariate $F$-matrix. Technical Report 84-89, Centre for Multivariate Analysis, Univ. Pittsburgh.
  • [6] Bai, Z.D. and Silverstein, J.W. (1999). Exact separation of eigenvalues of large-dimensional sample covariance matrices. Ann. Probab. 27 1536–1555.
  • [7] Bai, Z.D. and Silverstein, J.W. (2004). CLT for linear spectral statistics of large-dimensional sample covariance matrices. Ann. Probab. 32 553–605.
  • [8] Bai, Z.D. and Yin, Y.Q. (1993). Limit of the smallest eigenvalue of a large-dimensional sample covariance matrix. Ann. Probab. 21 1275–1294.
  • [9] Bai, Z.D., Yin, Y.Q. and Krishnaiah, P.R. (1986). On limiting spectral distribution of product of two random matrices when the underlying distribution is isotropic. J. Multivariate Anal. 19 189–200.
  • [10] Cai, T., Liu, W. and Xia, Y. (2013). Two-sample covariance matrix testing and support recovery in high-dimensional and sparse settings. J. Amer. Statist. Assoc. 108 265–277.
  • [11] Chatterjee, S. (2009). Fluctuations of eigenvalues and second order Poincaré inequalities. Probab. Theory Related Fields 143 1–40.
  • [12] Dempster, A.P. (1958). A high dimensional two sample significance test. Ann. Math. Stat. 29 995–1010.
  • [13] 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.
  • [14] Li, J. and Chen, S.X. (2012). Two sample tests for high-dimensional covariance matrices. Ann. Statist. 40 908–940.
  • [15] Pan, G.M. and Zhou, W. (2008). Central limit theorem for signal-to-interference ratio of reduced rank linear receiver. Ann. Appl. Probab. 18 1232–1270.
  • [16] Pillai, K.C.S. (1967). Upper percentage points of the largest root of a matrix in multivariate analysis. Biometrika 54 189–194.
  • [17] Pillai, K.C.S. and Flury, B.N. (1984). Percentage points of the largest characteristic root of the multivariate beta matrix. Comm. Statist. Theory Methods 13 2199–2237.
  • [18] Schott, J.R. (2007). Some high-dimensional tests for a one-way MANOVA. J. Multivariate Anal. 98 1825–1839.
  • [19] Schott, J.R. (2007). A test for the equality of covariance matrices when the dimension is large relative to the sample sizes. Comput. Statist. Data Anal. 51 6535–6542.
  • [20] Silverstein, J.W. (1985). The limiting eigenvalue distribution of a multivariate $F$ matrix. SIAM J. Math. Anal. 16 641–646.
  • [21] Silverstein, J.W. (1995). Strong convergence of the empirical distribution of eigenvalues of large-dimensional random matrices. J. Multivariate Anal. 55 331–339.
  • [22] Silverstein, J.W. and Choi, S.-I. (1995). Analysis of the limiting spectral distribution of large-dimensional random matrices. J. Multivariate Anal. 54 295–309.
  • [23] Srivastava, M.S. (2005). Some tests concerning the covariance matrix in high dimensional data. J. Japan Statist. Soc. 35 251–272.
  • [24] Srivastava, M.S., Kollo, T. and von Rosen, D. (2011). Some tests for the covariance matrix with fewer observations than the dimension under non-normality. J. Multivariate Anal. 102 1090–1103.
  • [25] Wachter, K.W. (1980). The limiting empirical measure of multiple discriminant ratios. Ann. Statist. 8 937–957.
  • [26] Yang, Y.R. and Pan, G.M. (2012). Independence test for high dimensional data based on regularized canonical correlation coefficients. Technical report.
  • [27] Yin, Y.Q., Bai, Z.D. and Krishnaiah, P.R. (1983). Limiting behavior of the eigenvalues of a multivariate $F$ matrix. J. Multivariate Anal. 13 508–516.
  • [28] Zheng, S. (2012). Central limit theorems for linear spectral statistics of large dimensional $F$-matrices. Ann. Inst. Henri Poincaré Probab. Stat. 48 444–476.
  • [29] Zheng, S., Bai, Z. and Yao, J. (2015). Substitution principle for CLT of linear spectral statistics of high-dimensional sample covariance matrices with applications to hypothesis testing. Ann. Statist. 43 546–591.
  • [30] Zheng, S.R., Bai, Z.D. and Yao, J.F. (2015). CLT for linear spectral statistics of a rescaled sample precision matrix. Random Matrix: Theory and Applications 4 1550014.