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. Although a standard loss function is the Stein loss, it is not available in the case of a high dimension. In this paper, a new type of a quadratic loss function, called the intrinsic loss, is suggested, and unified dominance results are derived under the 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. AMS 2010 subject classifications: Primary 62F11, 62J12, Secondary 62C15, 62C20.