## The Annals of Statistics

### Nonlinear shrinkage estimation of large-dimensional covariance matrices

#### Abstract

Many statistical applications require an estimate of a covariance matrix and/or its inverse. When the matrix dimension is large compared to the sample size, which happens frequently, the sample covariance matrix is known to perform poorly and may suffer from ill-conditioning. There already exists an extensive literature concerning improved estimators in such situations. In the absence of further knowledge about the structure of the true covariance matrix, the most successful approach so far, arguably, has been shrinkage estimation. Shrinking the sample covariance matrix to a multiple of the identity, by taking a weighted average of the two, turns out to be equivalent to linearly shrinking the sample eigenvalues to their grand mean, while retaining the sample eigenvectors. Our paper extends this approach by considering nonlinear transformations of the sample eigenvalues. We show how to construct an estimator that is asymptotically equivalent to an oracle estimator suggested in previous work. As demonstrated in extensive Monte Carlo simulations, the resulting bona fide estimator can result in sizeable improvements over the sample covariance matrix and also over linear shrinkage.

#### Article information

Source
Ann. Statist. Volume 40, Number 2 (2012), 1024-1060.

Dates
First available in Project Euclid: 18 July 2012

https://projecteuclid.org/euclid.aos/1342625460

Digital Object Identifier
doi:10.1214/12-AOS989

Mathematical Reviews number (MathSciNet)
MR2985942

Zentralblatt MATH identifier
1274.62371

Subjects
Primary: 62H12: Estimation
Secondary: 62G20: Asymptotic properties 15A52

#### Citation

Ledoit, Olivier; Wolf, Michael. Nonlinear shrinkage estimation of large-dimensional covariance matrices. Ann. Statist. 40 (2012), no. 2, 1024--1060. doi:10.1214/12-AOS989. https://projecteuclid.org/euclid.aos/1342625460

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