Abstract
We consider sample covariance matrices \({S_N=\frac{1}{p}\Sigma_N^{1/2}X_NX_N^* \Sigma_N^{1/2}}\) where X N is a N × p real or complex matrix with i.i.d. entries with finite 12th moment and Σ N is a N × N positive definite matrix. In addition we assume that the spectral measure of Σ N almost surely converges to some limiting probability distribution as N → ∞ and p/N → γ > 0. We quantify the relationship between sample and population eigenvectors by studying the asymptotics of functionals of the type \({\frac{1}{N}\text{Tr} ( g(\Sigma_N) (S_N-zI)^{-1}),}\) where I is the identity matrix, g is a bounded function and z is a complex number. This is then used to compute the asymptotically optimal bias correction for sample eigenvalues, paving the way for a new generation of improved estimators of the covariance matrix and its inverse.
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Bai Z.D.: Convergence rate of expected spectral distributions of large random matrices. ii, sample covariance matrices. Ann. Probab. 21, 649–672 (1993)
Bai Z.D.: Methodologies in spectral analysis of large dimensional random matrices, a review. Stat. Sin. 9, 611–677 (1999)
Bai Z.D., Miao B.Q., Pan G.M.: On asymptotics of eigenvectors of large sample covariance matrix. Ann. Probab. 35, 1532–1572 (2007)
Bai Z.D., Silverstein J.W.: On the empirical distribution of eigenvalues of a class of large dimensional random matrices. J. Multivariate Anal. 54, 175–192 (1995)
Bai Z.D., Silverstein J.W.: No eigenvalues outside the support of the limiting spectral distribution of large-dimensional sample covariance matrices. Ann. Probab. 26, 316–345 (1998)
Bai Z.D., Silverstein J.W.: Exact separation of eigenvalues of large-dimensional sample covariance matrices. Ann. Probab. 27, 1536–1555 (1999)
Bai Z.D., Silverstein J.W.: Clt for linear spectral statistics of large-dimensional sample covariance matrices. Ann. Probab. 32, 553–605 (2004)
Bai Z.D., Silverstein J.W.: Spectral Analysis of Large Dimensional Random Matrices. Science Press, Beijing (2006)
Bai Z.D., Yin Y.Q.: Limit of the smallest eigenvalue of a large-dimensional sample covariance matrix. Ann. Probab. 21, 1275–1294 (1993)
Bickel P.J., Levina E.: Regularized estimation of large covariance matrices. Ann. Stat. 36(1), 199–227 (2008)
Choi S.I., Silverstein J.W.: Analysis of the limiting spectral distribution of large dimensional random matrices. J. Multivariate Anal. 54, 295–309 (1995)
Combettes P.L., Silverstein J.W.: Signal detection via spectral theory of large dimensional random matrices. IEEE Trans. Signal Process. 40, 2100–2105 (1992)
El Karoui N.: Spectrum estimation for large dimensional covariance matrices using random matrix theory. Ann. Stat. 36, 2757–2790 (2008)
Grenander U., Silverstein J.W.: Spectral analysis of networks with random topologies. SIAM J. Appl. Math. 32, 499–519 (1977)
Johnstone I.M.: On the distribution of the largest eigenvalue in principal components analysis. Ann. Stat. 29, 295–327 (2001)
Krishnaiah P.R., Yin Y.Q.: A limit theorem for the eigenvalues of product of two random matrices. J. Multivariate Anal. 13, 489–507 (1983)
Ledoit O., Wolf M.: Honey, i shrunk the sample covariance matrix. J. Portfolio Manage. 30, 110–119 (2004)
Ledoit, O.: Essays on risk and return in the stock market. Ph.D. thesis, Massachusetts Institute of Technology, USA (1995)
Ledoit O., Wolf M.: Some hypothesis tests for the covariance matrix when the dimension is large compared to the sample size. Ann. Stat. 30, 1081–1102 (2002)
Ledoit O., Wolf M.: A well-conditioned estimator for large-dimensional covariance matrices. J. Multivariate Anal. 88, 365–411 (2004)
Marčenko V.A., Pastur L.A.: Distribution of eigenvalues for some sets of random matrices. Math. USSR-Sb. 1, 457–486 (1967)
Pan G.M., Zhou W.: Central limit theorem for signal-to-interference ratio of reduced rank linear receiver. Ann. Appl. Probab. 18, 1232–1270 (2008)
Peche S.: Universality results for the largest eigenvalues of some sample covariance matrix ensembles. Prob. Theor. Relat. Fields 143(3–4), 481–516 (2009)
Perlman, M.D.: Multivariate Statistical Analysis. Online textbook published by the Department of Statistics of the University of Washington, Seattle (2007)
Silverstein J.W.: Some limit theorems on the eigenvectors of large dimensional sample covariance matrices. J. Multivariate Anal. 15, 295–324 (1984)
Silverstein J.W.: On the eigenvectors of large dimensional sample covariance matrices. J. Multivariate Anal. 30, 1–16 (1989)
Silverstein J.W.: Weak convergence of random functions defined by the eigenvectors of sample covariance matrices. Ann. Probab. 18, 1174–1194 (1990)
Silverstein J.W.: Strong convergence of the empirical distribution of eigenvalues of large dimensional random matrices. J. Multivariate Anal. 55, 331–339 (1995)
Wachter K.W.: The strong limits of random matrix spectra for sample matrices of independent elements. Ann. Probab. 6, 1–18 (1978)
Yin Y.Q.: Limiting spectral distribution for a class of random matrices. J. Multivariate Anal. 20, 50–68 (1986)
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Ledoit, O., Péché, S. Eigenvectors of some large sample covariance matrix ensembles. Probab. Theory Relat. Fields 151, 233–264 (2011). https://doi.org/10.1007/s00440-010-0298-3
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DOI: https://doi.org/10.1007/s00440-010-0298-3
Keywords
- Asymptotic distribution
- Bias correction
- Eigenvectors and eigenvalues
- Principal component analysis
- Random matrix theory
- Sample covariance matrix
- Shrinkage estimator
- Stieltjes transform