Header

UZH-Logo

Maintenance Infos

Nonlinear Shrinkage Estimation of Large-Dimensional Covariance Matrices


Ledoit, Olivier; Wolf, Michael (2011). Nonlinear Shrinkage Estimation of Large-Dimensional Covariance Matrices. Working paper series / Institute for Empirical Research in Economics No. 515, University of Zurich.

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.

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.

Statistics

Downloads

176 downloads since deposited on 04 Jan 2012
12 downloads since 12 months
Detailed statistics

Additional indexing

Item Type:Working Paper
Communities & Collections:03 Faculty of Economics > Department of Economics
Working Paper Series > Institute for Empirical Research in Economics (former)
Dewey Decimal Classification:330 Economics
JEL Classification:C13
Uncontrolled Keywords:Large-dimensional asymptotics, nonlinear shrinkage, rotation equivariance
Language:English
Date:December 2011
Deposited On:04 Jan 2012 15:45
Last Modified:12 Aug 2017 14:37
Series Name:Working paper series / Institute for Empirical Research in Economics
Number of Pages:16
ISSN:1424-0459
Additional Information:Revised version
Official URL:http://www.iew.unizh.ch/wp/iewwp515.pdf
Related URLs:http://www.econ.uzh.ch/wp.html

Download

Preview Icon on Download
Preview
Filetype: PDF (Version 2011)
Size: 553kB
Preview Icon on Download
Preview
Filetype: PDF (Version 2010)
Size: 429kB