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Covariance regularization by thresholding. (English) Zbl 1196.62062
Summary: 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 \((\log p)/n\rightarrow 0\), 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.

MSC:
62H12 Estimation in multivariate analysis
62F12 Asymptotic properties of parametric estimators
62F40 Bootstrap, jackknife and other resampling methods
65C60 Computational problems in statistics (MSC2010)
62P12 Applications of statistics to environmental and related topics
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