This paper considers regularizing a covariance matrix of p variables estimated from n observations, by hard thresholding. We show that the thresholded estimate is consistent in the operator norm as long as the true covariance matrix is sparse in a suitable sense, the variables are Gaussian or sub-Gaussian, and (log p)/n→0, and obtain explicit rates. The results are uniform over families of covariance matrices which satisfy a fairly natural notion of sparsity. We discuss an intuitive resampling scheme for threshold selection and prove a general cross-validation result that justifies this approach. We also compare thresholding to other covariance estimators in simulations and on an example from climate data.
Publié le : 2008-12-15
Classification:
Covariance estimation,
regularization,
sparsity,
thresholding,
large p small n,
high dimension low sample size,
62H12,
62F12,
62G09
@article{1231165180,
author = {Bickel, Peter J. and Levina, Elizaveta},
title = {Covariance regularization by thresholding},
journal = {Ann. Statist.},
volume = {36},
number = {1},
year = {2008},
pages = { 2577-2604},
language = {en},
url = {http://dml.mathdoc.fr/item/1231165180}
}
Bickel, Peter J.; Levina, Elizaveta. Covariance regularization by thresholding. Ann. Statist., Tome 36 (2008) no. 1, pp. 2577-2604. http://gdmltest.u-ga.fr/item/1231165180/