# Spectrum estimation: a unified framework for covariance matrix estimation and PCA in large dimensions - Zurich Open Repository and Archive

Ledoit, Olivier; Wolf, Michael (2013). Spectrum estimation: a unified framework for covariance matrix estimation and PCA in large dimensions. Working paper series / Department of Economics 105, University of Zurich.

## 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.

## 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.

Detailed statistics

Item Type: Working Paper 03 Faculty of Economics > Department of Economics Working Paper Series > Department of Economics 330 Economics C13 Large-dimensional asymptotics, covariance matrix eigenvalues, nonlinear shrinkage, principal component analysis English July 2013 09 Jan 2013 08:24 06 Apr 2017 08:23 Working paper series / Department of Economics 40 1664-7041 Revised version http://www.econ.uzh.ch/static/wp/econwp105.pdf http://www.econ.uzh.ch/static/workingpapers.php

Filetype: PDF (Version January 2013) - Registered users only
Size: 438kB
Filetype: PDF (Revised version March 2013) - Registered users only
Size: 442kB
Preview
Filetype: PDF (Revised version July 2013)
Size: 557kB

## 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.