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Eigenvectors of some large sample covariance matrices ensembles

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  • Olivier Ledoit
  • Sandrine Péché

Abstract

We consider sample covariance matrices constructed from real or complex i.i.d. variates with finite 12th moment. We assume that the population covariance matrix is positive definite and its spectral measure almost surely converges to some limiting probability distribution as the number of variables and the number of observations go to infinity together, with their ratio converging to a finite positive limit. We quantify the relationship between sample and population eigenvectors, by studying the asymptotics of a broad family of functionals that generalizes the Stieltjes transform of the spectral measure. 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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Bibliographic Info

Paper provided by Institute for Empirical Research in Economics - University of Zurich in its series IEW - Working Papers with number 407.

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Date of creation: Mar 2009
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Handle: RePEc:zur:iewwpx:407

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Keywords: Asymptotic distribution; bias correction; eigenvectors and eigenvalues; principal component analysis; random matrix theory; sample covariance matrix; shrinkage estimator; Stieltjes transform.;

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  1. Olivier Ledoit & Michael Wolf, 2001. "Some hypothesis tests for the covariance matrix when the dimension is large compared to the sample size," Economics Working Papers 575, Department of Economics and Business, Universitat Pompeu Fabra.
  2. Silverstein, Jack W., 1984. "Some limit theorems on the eigenvectors of large dimensional sample covariance matrices," Journal of Multivariate Analysis, Elsevier, vol. 15(3), pages 295-324, December.
  3. Silverstein, J. W. & Bai, Z. D., 1995. "On the Empirical Distribution of Eigenvalues of a Class of Large Dimensional Random Matrices," Journal of Multivariate Analysis, Elsevier, vol. 54(2), pages 175-192, August.
  4. Silverstein, J. W., 1995. "Strong Convergence of the Empirical Distribution of Eigenvalues of Large Dimensional Random Matrices," Journal of Multivariate Analysis, Elsevier, vol. 55(2), pages 331-339, November.
  5. Yin, Y. Q., 1986. "Limiting spectral distribution for a class of random matrices," Journal of Multivariate Analysis, Elsevier, vol. 20(1), pages 50-68, October.
  6. Yin, Y. Q. & Krishnaiah, P. R., 1983. "A limit theorem for the eigenvalues of product of two random matrices," Journal of Multivariate Analysis, Elsevier, vol. 13(4), pages 489-507, December.
  7. Ledoit, Olivier & Wolf, Michael, 2004. "A well-conditioned estimator for large-dimensional covariance matrices," Journal of Multivariate Analysis, Elsevier, vol. 88(2), pages 365-411, February.
  8. Silverstein, J. W. & Choi, S. I., 1995. "Analysis of the Limiting Spectral Distribution of Large Dimensional Random Matrices," Journal of Multivariate Analysis, Elsevier, vol. 54(2), pages 295-309, August.
  9. Silverstein, Jack W., 1989. "On the eigenvectors of large dimensional sample covariance matrices," Journal of Multivariate Analysis, Elsevier, vol. 30(1), pages 1-16, July.
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