AISM 53, 760-768
© 2001 ISM

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,
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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