Improved estimation of the covariance matrix and the generalized variance of a multivariate normal distribution: some unifying results

摘要

Suppose that there is a typical (scale equivariant) improved estimator of the variance, σ~2, of a univariate normal distribution with unknown mean at our disposal. Using this estimator, in this work we construct in a very simple way an improved estimator of the covariance matrix, Σ, of a multivariate normal distribution with unknown mean and another improved estimator of the generalized variance, |Σ|. The data is a sample of i.i.d. observations from this distribution. The loss function is the entropy loss or the quadratic loss in the case of Σ and a general loss satisfying a certain condition in the case of |Σ|. These novel results reduce, in a specific way, the problems of estimating Σ or |Σ| to the univariate problem of estimating σ~2. As a consequence, Stein-type, Brewster and Zidek-type, Strawderman-type and Maruyama-type improved estimators for Σ and |Σ| are directly constructed from their univariate counterparts. This work unifies and extends previously obtained results on the estimation of Σ and |Σ|.