NOVELIST estimator of large correlation and covariance matrices and their inverses

Abstract

We propose a “NOVEL Integration of the Sample and Thresholded covariance” (NOVELIST) estimator to estimate the large covariance (correlation) and precision matrix. NOVELIST estimator performs shrinkage of the sample covariance (correlation) towards its thresholded version. The sample covariance (correlation) component is non-sparse and can be low rank in high dimensions. The thresholded sample covariance (correlation) component is sparse, and its addition ensures the stable invertibility of NOVELIST. The benefits of the NOVELIST estimator include simplicity, ease of implementation, computational efficiency and the fact that its application avoids eigenanalysis. We obtain an explicit convergence rate in the operator norm over a large class of covariance (correlation) matrices when the dimension p and the sample size n satisfy log p/ n→ 0 , and its improved version when p/ n→ 0. In empirical comparisons with several popular estimators, the NOVELIST estimator performs well in estimating covariance and precision matrices over a wide range of models and sparsity classes. Real-data applications are presented.