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Positive-definite \(\ell_1\)-penalized estimation of large covariance matrices. (English) Zbl 1258.62063
Summary: 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 \(\ell_1\)-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, non-asymptotic statistical theory is also established for the proposed estimator. The competitive finite-sample performance of our proposal is demonstrated by both simulations and real applications.

MSC:
62H12 Estimation in multivariate analysis
62F12 Asymptotic properties of parametric estimators
65C60 Computational problems in statistics (MSC2010)
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