On convergence of regularized covariance estimator based on modified Cholesky decomposition

The regularization for covariance matrix is a widely used technique when estimating large covariance matrices. This paper examines a penalized likelihood method for constructing a statistically efficient covariance matrix estimator. Modified Cholesky decomposition (MCD) is used to parameterize the covariance matrix and the effective regularization scheme is achieved by combining both shrinkage and smoothing penalties on the Cholesky factor. The practical performance is at odds with an absence of theoretical properties of the derived estimators in the literature. In this work, we aim to fill the gap between theory and practice by establishing the convergence properties under regularity conditions. We also provide a simulation study as numerical illustrations.