Header

UZH-Logo

Maintenance Infos

Nonlinear shrinkage estimation of large-dimensional covariance matrices


Ledoit, Olivier; Wolf, Michael (2012). Nonlinear shrinkage estimation of large-dimensional covariance matrices. The Annals of Statistics, 40(2):1024-1060.

Abstract

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

Abstract

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

Statistics

Citations

43 citations in Web of Science®
29 citations in Scopus®
Google Scholar™

Altmetrics

Downloads

21 downloads since deposited on 17 Sep 2012
1 download since 12 months
Detailed statistics

Additional indexing

Item Type:Journal Article, refereed, original work
Communities & Collections:03 Faculty of Economics > Department of Economics
Dewey Decimal Classification:330 Economics
Language:English
Date:2012
Deposited On:17 Sep 2012 08:49
Last Modified:05 Apr 2016 15:57
Publisher:Institute of Mathematical Statistics
ISSN:0090-5364
Free access at:Publisher DOI. An embargo period may apply.
Publisher DOI:https://doi.org/10.1214/12-AOS989
Other Identification Number:merlin-id:7249

Download

Preview Icon on Download
Content: Submitted Version
Filetype: PDF - Registered users only
Size: 553kB
View at publisher
Preview Icon on Download
Preview
Content: Published Version
Filetype: PDF
Size: 589kB

TrendTerms

TrendTerms displays relevant terms of the abstract of this publication and related documents on a map. The terms and their relations were extracted from ZORA using word statistics. Their timelines are taken from ZORA as well. The bubble size of a term is proportional to the number of documents where the term occurs. Red, orange, yellow and green colors are used for terms that occur in the current document; red indicates high interlinkedness of a term with other terms, orange, yellow and green decreasing interlinkedness. Blue is used for terms that have a relation with the terms in this document, but occur in other documents.
You can navigate and zoom the map. Mouse-hovering a term displays its timeline, clicking it yields the associated documents.

Author Collaborations