Ensemble Estimation of Large Sparse Covariance Matrix Based on Modified Cholesky Decomposition

Estimation of large sparse covariance matrices is of great importance for statistical analysis, especially in the high dimensional setting. The traditional approach such as sample covariance matrix could perform poorly due to the high dimensionality. In this work, we propose a positive-definite estimator for the covariance matrix based on the modified Cholesky decomposition. The modified Cholesky decomposition relies on the order of variables, which provides the flexibility to obtain a set of covariance matrix estimates under different orders of variables. The proposed method considers an ensemble estimator as the center of such a set of covariance matrix estimates with respect to the Frobenius norm. The proposed estimator is not only guaranteed to be positive definite, but also can capture the underlying sparse structure of the covariance matrix. Under some weak regularity conditions, both algorithmic convergence and asymptotical convergence are established. The merits of the proposed method are illustrated through simulation studies and one real data example.