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High dimensional covariance matrix estimation by penalizing the matrix-logarithm transformed likelihood

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  • Yu, Philip L.H.
  • Wang, Xiaohang
  • Zhu, Yuanyuan

Abstract

It is well known that when the dimension of the data becomes very large, the sample covariance matrix S will not be a good estimator of the population covariance matrix Σ. Using such estimator, one typical consequence is that the estimated eigenvalues from S will be distorted. Many existing methods tried to solve the problem, and examples of which include regularizing Σ by thresholding or banding. In this paper, we estimate Σ by maximizing the likelihood using a new penalization on the matrix logarithm of Σ (denoted by A) of the form: ‖A−mI‖F2=∑i(log(di)−m)2, where di is the ith eigenvalue of Σ. This penalty aims at shrinking the estimated eigenvalues of A toward the mean eigenvalue m. The merits of our method are that it guarantees Σ to be non-negative definite and is computational efficient. The simulation study and applications on portfolio optimization and classification of genomic data show that the proposed method outperforms existing methods.

Suggested Citation

  • Yu, Philip L.H. & Wang, Xiaohang & Zhu, Yuanyuan, 2017. "High dimensional covariance matrix estimation by penalizing the matrix-logarithm transformed likelihood," Computational Statistics & Data Analysis, Elsevier, vol. 114(C), pages 12-25.
    • Handle: RePEc:eee:csdana:v:114:y:2017:i:c:p:12-25
    DOI: 10.1016/j.csda.2017.04.004
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    References listed on IDEAS

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    1. Javier Ibáñez & Jorge Sastre & Pedro Ruiz & José M. Alonso & Emilio Defez, 2021. "An Improved Taylor Algorithm for Computing the Matrix Logarithm," Mathematics, MDPI, vol. 9(17), pages 1-19, August.

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