A Best Linear Empirical Bayes Method for High-Dimensional Covariance Matrix Estimation

Abstract

Covariance matrix estimation plays a significant role in both in the theory and practice of portfolio analysis and risk management. This paper deals with the available data prior to developing a factor model to enhance covariance matrix estimation. Our work has two main outcomes. First, for a general linear model with unknown prior parameters, a class of best linear empirical Bayes estimators is established through two kinds of architectures to improve the estimation accuracy by utilizing additional data prior. The theoretical results indicate two key points: the proposed estimators are equivalent to the linear minimum mean-square error estimator when complete or sufficient partial data prior are provided; and the proposed estimators perform better than the optimal weighted least squares method, which ignores the data prior in each situation. Second, the proposed estimators are used for calculating a high-dimensional covariance matrix through factor models. The numerical example and the simulation results verify the effectiveness of our methods.

Sparse precision matrix estimation under lower polynomial moment assumption

“…In recent years, various regularization methods have been proposed. For example, l 1 -penalized normal likelihood method [7,8] maximum likelihood method [9][10][11], penalized generalized Sylvester matrix equation [12], Dantzig selector method [13] and modified Cholesky's decomposition-based method [14,15], and among others.…”