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December 2008 Operator norm consistent estimation of large-dimensional sparse covariance matrices
Noureddine El Karoui
Ann. Statist. 36(6): 2717-2756 (December 2008). DOI: 10.1214/07-AOS559


Estimating covariance matrices is a problem of fundamental importance in multivariate statistics. In practice it is increasingly frequent to work with data matrices X of dimension n×p, where p and n are both large. Results from random matrix theory show very clearly that in this setting, standard estimators like the sample covariance matrix perform in general very poorly.

In this “large n, large p” setting, it is sometimes the case that practitioners are willing to assume that many elements of the population covariance matrix are equal to 0, and hence this matrix is sparse. We develop an estimator to handle this situation. The estimator is shown to be consistent in operator norm, when, for instance, we have pn as n→∞. In other words the largest singular value of the difference between the estimator and the population covariance matrix goes to zero. This implies consistency of all the eigenvalues and consistency of eigenspaces associated to isolated eigenvalues.

We also propose a notion of sparsity for matrices, that is, “compatible” with spectral analysis and is independent of the ordering of the variables.


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Noureddine El Karoui. "Operator norm consistent estimation of large-dimensional sparse covariance matrices." Ann. Statist. 36 (6) 2717 - 2756, December 2008.


Published: December 2008
First available in Project Euclid: 5 January 2009

zbMATH: 1196.62064
MathSciNet: MR2485011
Digital Object Identifier: 10.1214/07-AOS559

Primary: 62H12

Keywords: adjacency matrices , correlation matrices , Covariance matrices , eigenvalues of covariance matrices , high-dimensional inference , multivariate statistical analysis , Random matrix theory , Sparsity , β-sparsity

Rights: Copyright © 2008 Institute of Mathematical Statistics


Vol.36 • No. 6 • December 2008
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