Positive-Definite ℓ₁-Penalized Estimation of Large Covariance Matrices

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 ℓ₁-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.

Keywords:

• Alternating direction methods
• Matrix norm
• Positive-definite estimation
• Soft-thresholding
• Sparsity.

Acknowledgments

The article was completed when Lingzhou Xue was a Ph.D. student at the University of Minnesota and Shiqian Ma was a Postdoctoral Fellow in the Institute for Mathematics and Its Applications at the University of Minnesota. The authors thank Adam Rothman for sharing his code. We are grateful to the coeditor, the associate editor, and two referees for their helpful and constructive comments. Shiqian Ma was supported by the National Science Foundation postdoctoral fellowship through the Institute for Mathematics and Its Applications at the University of Minnesota. Lingzhou Xue and Hui Zou are supported in part by grants from the National Science Foundation and the Office of Naval Research.

Notes

NOTE: Each metric is averaged over 100 replications with the standard error shown in the bracket. NA means that the results for (Rothman’s method) are not available due to the extremely long run times.

NOTE: Each metric is averaged over 100 replications with the standard error shown in the bracket. NA means that the results for (Rothman’s method) are not available due to the extremely long run times.

NOTE: Monthly mean returns, standard deviations of monthly returns, and corresponding Sharpe ratios are all expressed in %.