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

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.