AISM 53, 760-768
© 2001 ISM
(Received May 12, 1999; revised November 15, 1999)
Abstract. Let $X_1,\ldots, X_n (n>p)$ be a random sample from multivariate normal distribution $N_p(\mu,\Sigma)$, where $\mu\in R^p$ and $\Sigma$ is a positive definite matrix, both $\mu$ and $\Sigma$ being unknown. We consider the problem of estimating the precision matrix $\Sigma^{-1}$. In this paper it is shown that for the entropy loss, the best lower-triangular affine equivariant minimax estimator of $\Sigma^{-1}$ is inadmissible and an improved estimator is explicitly constructed. Note that our improved estimator is obtained from the class of lower-triangular scale equivariant estimators.
Key words and phrases: Best lower-triangular equivariant minimax estimator, precision matrix, inadmissibility, multivariate normal distribution, risk function, the entropy loss.