Xinwei Deng and Kam-Wah Tsui
University of Wisconsin-Madison
Penalized Covariance Matrix Estimation using a Matrix-Logarithm Transformation
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.