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


  • Lingzhou Xue
  • Shiqian Ma
  • Hui Zou


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.

Suggested Citation

  • Lingzhou Xue & Shiqian Ma & Hui Zou, 2012. "Positive-Definite ℓ 1 -Penalized Estimation of Large Covariance Matrices," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 107(500), pages 1480-1491, December.
  • Handle: RePEc:taf:jnlasa:v:107:y:2012:i:500:p:1480-1491
    DOI: 10.1080/01621459.2012.725386

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    Cited by:

    1. Avagyan, Vahe & Alonso, Andrés M. & Nogales, Francisco J., 2015. "D-trace Precision Matrix Estimation Using Adaptive Lasso Penalties," DES - Working Papers. Statistics and Econometrics. WS 21775, Universidad Carlos III de Madrid. Departamento de Estadística.
    2. Cui, Ying & Leng, Chenlei & Sun, Defeng, 2016. "Sparse estimation of high-dimensional correlation matrices," Computational Statistics & Data Analysis, Elsevier, vol. 93(C), pages 390-403.
    3. Jianqing Fan & Yuan Liao & Martina Mincheva, 2013. "Large covariance estimation by thresholding principal orthogonal complements," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 75(4), pages 603-680, September.
    4. Kenneth Lange & Eric C. Chi & Hua Zhou, 2014. "A Brief Survey of Modern Optimization for Statisticians," International Statistical Review, International Statistical Institute, vol. 82(1), pages 46-70, April.
    5. Denis Belomestny & Mathias Trabs & Alexandre Tsybakov, 2017. "Sparse covariance matrix estimation in high-dimensional deconvolution," Working Papers 2017-25, Center for Research in Economics and Statistics.
    6. Binyan Jiang, 2015. "An empirical estimator for the sparsity of a large covariance matrix under multivariate normal assumptions," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 67(2), pages 211-227, April.
    7. Zou, Tao & Chen, Song Xi, 2014. "Enhancing Estimation for Interest Rate Diffusion Models with Bond Prices," MPRA Paper 67073, University Library of Munich, Germany, revised Apr 2015.
    8. repec:eee:csdana:v:114:y:2017:i:c:p:12-25 is not listed on IDEAS
    9. Bai, Jushan & Liao, Yuan, 2016. "Efficient estimation of approximate factor models via penalized maximum likelihood," Journal of Econometrics, Elsevier, vol. 191(1), pages 1-18.

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