Improving on the Best Affine Equivariant Estimator
of the Ratio of Generalized Variances
George Iliopoulos and Stavros Kourouklis
We consider the problem of decision-theoretic estimation of the ratio of
generalized variances of two matrix normal distrbutions
with unknown means under a general loss function. The inadmissibility of the
best affine equivariant estimator is established by
exhibiting various improved estimators. In particular, under certain conditions
on the loss, two classes of improved procedures based on all the available data
are presented. As a preliminary result of independent interest, an improved
estimator of an arbitrary power of the generalized variance of a matrix normal
distribution with an unknown mean is derived under a general strictly
bowl-shaped loss.
AMS 1991 subject classifications: 62C99, 62H12, 62F10.
Key words and phrases: Equivariant estimation,
Stein technique, Brewster and Zidek tehnique, matrix normal distribution, Wishart
distribution, generalized variance, monotone likelihood ratio.