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Positive-Definite ℓ 1 -Penalized Estimation of Large Covariance Matrices

Lingzhou Xue, Shiqian Ma and Hui Zou

Journal of the American Statistical Association, 2012, vol. 107, issue 500, 1480-1491

Abstract: The thresholding covariance estimator has nice asymptotic properties for estimating sparse large covariance matrices, but it often has negative eigenvalues when used in real data analysis. To fix this drawback of thresholding estimation, we develop a positive-definite ℓ 1 -penalized covariance estimator for estimating sparse large covariance matrices. We derive an efficient alternating direction method to solve the challenging optimization problem and establish its convergence properties. Under weak regularity conditions, nonasymptotic statistical theory is also established for the proposed estimator. The competitive finite-sample performance of our proposal is demonstrated by both simulation and real applications.

Date: 2012
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