Estimation and optimal structure selection of high-dimensional Toeplitz covariance matrix

The estimation of structured covariance matrix arises in many fields. An appropriate covariance structure not only improves the accuracy of covariance estimation but also increases the efficiency of mean parameter estimators in statistical models. In this paper, a novel statistical method is proposed, which selects the optimal Toeplitz covariance structure and estimates the covariance matrix, simultaneously. An entropy loss function with nonconvex penalty is employed as a matrix-discrepancy measure, under which the optimal selection of sparse or nearly sparse Toeplitz structure and the parameter estimators of covariance matrix are made, simultaneously, through its minimization. The cases of both low-dimensional ( p ≤ n ) and high-dimensional ( p > n ) covariance matrix estimation are considered. The resulting Toeplitz structured covariance estimators are guaranteed to be positive definite and consistent. Asymptotic properties are investigated and simulation studies are conducted, showing that very high accurate Toeplitz covariance structure estimation is made. The proposed method is then applied to practical data analysis, which demonstrates its good performance in covariance estimation in practice.