Estimation of Large Covariance Matrices by Shrinking to Structured Target in Normal and Non-Normal Distributions

Abstract

This paper addresses the estimation of large-dimensional covariance matrices under both normal and nonnormal distributions. The shrinkage estimators are constructed by convexly combining the sample covariance matrix and a structured target matrix. The optimal oracle shrinkage intensity is obtained analytically for any prespecified target in a class of matrices which includes various structured matrices such as banding, thresholding, diagonal, and block diagonal matrices. After deriving the unbiased and consistent estimates of some quantities in the oracle intensity involving unknown population covariance matrix, two classes of available optimal intensities are proposed under normality and nonnormality, respectively, by plug-in technique. For the target matrix with unknown parameter such as bandwidth in banded target, an analytic estimate of unknown parameter is provided. Both the numerical simulations and applications to signal processing and discriminant analysis show the comparable performance of the proposed estimators for large-dimensional data.