ESTIMATION OF THE MULTIVARIATE NORMAL PRECISION MATRIX UNDER THE ENTROPY LOSS

XIAN ZHOU^{1}, XIAOQIAN SUN^{2} AND JINGLONG WANG^{2}

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}, …, X_{n}(n > p) be a random sample from multivariate normal distribution N_{p}(μ, Σ), where μ ∈ R^{p} and Σ is a positive definite matrix, both μ and Σ being unknown. We consider the problem of estimating the precision matrix Σ^{−1}. In this paper it is shown that for the entropy loss, the best lower-triangular affine equivariant minimax estimator of Σ^{−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.