Unified improvements in estimation of a normal covariance matrix in high and low dimensions

The problem of estimating a covariance matrix in multivariate linear regression models is addressed in a decision-theoretic framework. This paper derives unified dominance results under a Stein-like loss, irrespective of order of the dimension, the sample size and the rank of the regression coefficients matrix. Especially, using the Stein–Haff identity, we develop a key inequality which is useful for constructing a truncated and improved estimator based on the information contained in the sample means or the ordinary least squares estimator of the regression coefficients. Also, a quadratic loss-like function is used to suggest alternative improved estimators with respect to an invariant quadratic loss.