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.

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.

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## Additional indexing

Item Type: | Working Paper |
---|---|

Communities & Collections: | 03 Faculty of Economics > Department of Economics
Working Paper Series > Department of Economics |

Dewey Decimal Classification: | 330 Economics |

JEL Classification: | C13 |

Uncontrolled Keywords: | Large-dimensional asymptotics, covariance matrix eigenvalues, nonlinear shrinkage, principal component analysis |

Language: | English |

Date: | July 2013 |

Deposited On: | 09 Jan 2013 08:24 |

Last Modified: | 05 Apr 2016 16:18 |

Series Name: | Working paper series / Department of Economics |

Number of Pages: | 40 |

ISSN: | 1664-7041 |

Additional Information: | Revised version |

Official URL: | http://www.econ.uzh.ch/static/wp/econwp105.pdf |

Related URLs: | http://www.econ.uzh.ch/static/workingpapers.php |

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