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Optimal estimation of a large-dimensional covariance matrix under Stein’s loss

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  • Olivier Ledoit
  • Michael Wolf

Abstract

This paper introduces a new method for deriving covariance matrix estimators that are decision-theoretically optimal. The key is to employ large-dimensional asymptotics: the matrix dimension and the sample size go to infinity together, with their ratio converging to a finite, nonzero limit. As the main focus, we apply this method to Stein’s loss. Compared to the estimator of Stein (1975, 1986), ours has five theoretical advantages: 1. it asymptotically minimizes the loss itself, instead of an estimator of the expected loss; 2. it does not necessitate post-processing through an ad hoc algorithm (called “isotonization”) to restore the positivity or the ordering of the covariance matrix eigenvalues; 3. it does not ignore any terms in the function to be minimized; 4. it does not require normality; and 5. it is not limited to applications where the sample size exceeds the dimension. In addition to these theoretical advantages, our estimator also improves upon Stein’ estimator in terms of finite-sample performance, as evidenced via extensive Monte Carlo simulations. To further demonstrate the effectiveness of our method, we show that some previously suggested estimators of the covariance matrix and its inverse are decision-theoretically optimal with respect to the Frobenius loss function.

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Bibliographic Info

Paper provided by Department of Economics - University of Zurich in its series ECON - Working Papers with number 122.

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Date of creation: May 2013
Date of revision: Dec 2013
Handle: RePEc:zur:econwp:122

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Related research

Keywords: Large-dimensional asymptotics; nonlinear shrinkage estimation; random matrix theory; rotation equivariance; Stein’s loss;

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  1. 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.
  2. 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.
  3. 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.
  4. 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.
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Cited by:
  1. Olivier Ledoit & Michael Wolf, 2014. "Nonlinear shrinkage of the covariance matrix for portfolio selection: Markowitz meets Goldilocks," ECON - Working Papers 137, Department of Economics - University of Zurich.

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