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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.

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*Tohru Okuzono *

2001-07-12