Adaptive ℓq regularized estimation for high-dimensional sparse covariance matrix

Abstract

The estimation of high-dimensional covariance matrix plays an important role in many application fields such as economics, biology, social and health sciences. A mainstream structural assumption for enhancing estimator accuracy is that the covariance matrix is sparse or approximately sparse. This paper proposes an adaptive ${\ell }_q(0<q<1)$ regularized estimator with minimum eigenvalue constraint for high-dimensional sparse covariance matrix. This method eliminates the need for the conventional two-stage framework of sequential correlation and covariance matrix estimation. Under appropriate regularity conditions, we analyze its asymptotic and finite sample properties. The proposed iterative reweighted minimization method and its inexact variant can be employed to find a desired estimate. Simulation studies confirm that the proposed estimation performs better than some other state-of-the-art methods.