Improvement on the Best Affine Equivariant Estimation of the Covariance Matrix under the Entropy Loss

Let X₁,…, X_n （n＞p） be a random sample from multivariate normal distribution N_p（μ, ∑）, where μ ∈R^P and ∑ is a positive definite matrix, both μ and ∑ being unknown. In this paper it is shown for the entropy loss L(\widehat\Sigma, \Sigma)=\operatornametr\left(\Sigma^-1 \widehat\Sigma\right)-\log \left|\Sigma^-1 \widehat\Sigma\right|-p the best affine equivariant estimator of the covariance matrix ∑ is inadmissible and an improved estimator is explicitly constructed.