The Annals of Statistics

Operator norm consistent estimation of large-dimensional sparse covariance matrices

Noureddine El Karoui

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Abstract

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.

Article information

Source
Ann. Statist. Volume 36, Number 6 (2008), 2717-2756.

Dates
First available in Project Euclid: 5 January 2009

Permanent link to this document
http://projecteuclid.org/euclid.aos/1231165183

Digital Object Identifier
doi:10.1214/07-AOS559

Mathematical Reviews number (MathSciNet)
MR2485011

Zentralblatt MATH identifier
05503374

Subjects
Primary: 62H12: Estimation

Keywords
Covariance matrices correlation matrices adjacency matrices eigenvalues of covariance matrices multivariate statistical analysis high-dimensional inference random matrix theory sparsity β-sparsity

Citation

El Karoui, Noureddine. Operator norm consistent estimation of large-dimensional sparse covariance matrices. Ann. Statist. 36 (2008), no. 6, 2717--2756. doi:10.1214/07-AOS559. http://projecteuclid.org/euclid.aos/1231165183.


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