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

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  • Olivier Ledoit
  • Michael Wolf

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.

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Bibliographic Info

Paper provided by Department of Economics - University of Zurich in its series ECON - Working Papers with number 105.

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Date of creation: Jan 2013
Date of revision: Jul 2013
Handle: RePEc:zur:econwp:105

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Keywords: Large-dimensional asymptotics; covariance matrix eigenvalues; nonlinear shrinkage; principal component analysis;

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Cited by:
  1. Olivier Ledoit & Michael Wolf, 2014. "Nonlinear shrinkage of the covariance matrix for portfolio selection: Markowitz meets Goldilocks," ECON - Working Papers 137, Department of Economics - University of Zurich.

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