**Xinwei Deng and Kam-Wah Tsui **

University of Wisconsin-Madison

For statistical inferences that involve covariance matrices, it is desirable to obtain an accurate covariance matrix estimate with a well-structured eigen-system. We propose to estimate the covariance matrix through its matrix logarithm based on an approximate log-likelihood function. We develop a generalization of the Leonard and Hsu (1992) log-likelihood approximation that no longer requires a nonsingular sample covariance matrix. The matrix log-transformation provides the ability to impose a convex penalty on the transformed likelihood such that the largest and smallest eigenvalues of the covariance matrix estimate can be regularized simultaneously. The proposed method transforms the problem of estimating the covariance matrix into the problem of estimating a symmetric matrix, which can be solved efficiently by an iterative quadratic programming algorithm. The performance of the proposed method is compared with other covariance matrix estimates through some simulation study and real applications.