Estimation of the noncentrality matrix of a noncentral Wishart distribution with unit scale matrix, employing a matrix loss function

Heinz Neudecker (University of Amsterdam)

Consider . The habitual unbiased estimator of is . Under certain conditions is better than , for suitable . Leung (1994) showed this using the loss function

We shall use a matrix loss function

and apply Lywner partial ordering of symmetric matrices. An approximate domination result will be proved, the error term being of order . We shall use a matrix version of a Fundamental Identity for the noncentral Wishart distribution. [Leung gave a scalar version extending Hass's Fundamental Identity (scalar version) for the central Wishart distribution.] A matrix version of Leung's ancillary Lemma 3.1 will then be established. We shall employ an approximation of being the expectation operator. A lemma of the matrix Hessian , where is a scalar (matrix) function of will be proved. Further a lemma on the scalar Hessian tr , where and are matrix functions of and is a constant matrix, will be given. References: Hass, L. R. (1981) Canadian J. Statist, 215-224. Leung, P. L. (1994) J. Multivariate Anal. 48, 107-14.