Journal
ANNALS OF STATISTICS
Volume 36, Issue 6, Pages 2577-2604Publisher
INST MATHEMATICAL STATISTICS-IMS
DOI: 10.1214/08-AOS600
Keywords
Covariance estimation; regularization; sparsity; thresholding; large p small n; high dimension low sample size
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Funding
- NSF [DMS-06-05236, DMS-05-05424, DMS-08-05798]
- NSA [MSPF-04Y-120]
- Division Of Mathematical Sciences
- Direct For Mathematical & Physical Scien [0805798] Funding Source: National Science Foundation
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This paper considers regularizing a covariance matrix of p variables estimated from it 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.
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