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


Ledoit, Olivier; Wolf, Michael (2018). Optimal estimation of a large-dimensional covariance matrix under Stein’s loss. Bernoulli, 24(4B):3791-3832.

Abstract

This paper introduces a new method for deriving covariance matrix estimators that are decision-theoretically optimal within a class of nonlinear shrinkage estimators. 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 (Estimation of a covariance matrix (1975); J. Math. Sci. 34 (1986) 1373–1403), 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 via 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’s 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 in the large-dimensional asymptotic limit with respect to the Frobenius loss function.

Abstract

This paper introduces a new method for deriving covariance matrix estimators that are decision-theoretically optimal within a class of nonlinear shrinkage estimators. 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 (Estimation of a covariance matrix (1975); J. Math. Sci. 34 (1986) 1373–1403), 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 via 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’s 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 in the large-dimensional asymptotic limit with respect to the Frobenius loss function.

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Additional indexing

Item Type:Journal Article, refereed, original work
Communities & Collections:03 Faculty of Economics > Department of Economics
Dewey Decimal Classification:330 Economics
Uncontrolled Keywords:large-dimensional asymptotics, nonlinear shrinkage estimation, random matrix theory, rotation equivariance, Stein’s loss
Language:English
Date:1 November 2018
Deposited On:15 Jan 2019 12:20
Last Modified:26 Apr 2019 17:56
Publisher:International Statistical Institute
ISSN:1350-7265
OA Status:Green
Free access at:Publisher DOI. An embargo period may apply.
Publisher DOI:https://doi.org/10.3150/17-bej979
Official URL:https://projecteuclid.org/download/pdfview_1/euclid.bj/1524038770

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