# Spectrum estimation: a unified framework for covariance matrix estimation and PCA in large dimensions

Ledoit, Olivier; Wolf, Michael (2015). Spectrum estimation: a unified framework for covariance matrix estimation and PCA in large dimensions. Journal of Multivariate Analysis, 139:360-384.

## Abstract

Covariance matrix estimation and principal component analysis (PCA) are two cornerstones of multivariate analysis. Classic textbook solutions perform poorly when the dimension of the data is of a magnitude similar to the sample size, or even larger. In such settings, there is a common remedy for both statistical problems: nonlinear shrinkage of the eigenvalues of the sample covariance matrix. The optimal nonlinear shrinkage formula depends on unknown population quantities and is thus not available. It is, however, possible to consistently estimate an oracle nonlinear shrinkage, which is motivated on asymptotic grounds. A key tool to this end is consistent estimation of the set of eigenvalues of the population covariance matrix (also known as the spectrum), an interesting and challenging
problem in its own right. Extensive Monte Carlo simulations demonstrate that our methods have desirable finite-sample properties and outperform previous proposals.

Covariance matrix estimation and principal component analysis (PCA) are two cornerstones of multivariate analysis. Classic textbook solutions perform poorly when the dimension of the data is of a magnitude similar to the sample size, or even larger. In such settings, there is a common remedy for both statistical problems: nonlinear shrinkage of the eigenvalues of the sample covariance matrix. The optimal nonlinear shrinkage formula depends on unknown population quantities and is thus not available. It is, however, possible to consistently estimate an oracle nonlinear shrinkage, which is motivated on asymptotic grounds. A key tool to this end is consistent estimation of the set of eigenvalues of the population covariance matrix (also known as the spectrum), an interesting and challenging
problem in its own right. Extensive Monte Carlo simulations demonstrate that our methods have desirable finite-sample properties and outperform previous proposals.

## Citations

3 citations in Web of Science®
1 citation in Scopus®

## Altmetrics

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Item Type: Journal Article, refereed, original work 03 Faculty of Economics > Department of Economics 330 Economics Large-dimensional asymptotics, covariance matrix eigenvalues, nonlinear shrinkage, principal component analysis English July 2015 10 May 2016 12:30 11 May 2016 03:07 Elsevier 0047-259X Siehe auch ECON Working Paper No. 105 unter http://www.zora.uzh.ch/70168/ Publisher DOI. An embargo period may apply. https://doi.org/10.1016/j.jmva.2015.04.006
Permanent URL: https://doi.org/10.5167/uzh-123999

Content: Submitted Version
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Embargo till: 2017-07-01

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