AISM 53, 760-768

## Estimation of the multivariate normal precision matrix under the entropy loss

### Xian Zhou1, Xiaoqian Sun2 and Jinglong Wang2

1Department of Applied Mathematics, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong, China, e-mail:maxzhou@polyu.edu.hk
2Department of Statistics, East China Normal University, Shanghai 200062, China

(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.

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