We consider the problem of decision-theoretic estimation of the ratio of generalized variances of two matrix normal distributions 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. Academic PressAMS 1991 subject classifications: 62C99, 62H12, 62F10.

The task of estimating the vector of parameters p in the general Gauss-Markov model (y, Xp, u2W) with no restrictions on the design matrix X or the covariance matrix rr2W is formulated as a constrained linear least squares problem. A I3LUE of any estimable function of p is obtained directly by solving this problem. The use of matrix decompositions leads to numerically stable algorithms for computing the solution. The approach is theoretically easy and is shown to be computationally more sound than methods based on generalized inverses. Practical expressions for the desired estimators, their covariance matrices, and an estimator of u2 are given.

At several places in the literature there are indications that many tests are optimal in the sense of Hodges-Lehmann efficiency. It is argued here that shrinkage of the acceptance regions of the tests to the null set in a coarse way is already enough to ensure optimality.This type of argument can be used to show optimality of e.g. Kolmogorov-Smirnov tests, Cram&-von Mises tests, and likelihood ratio tests and many other tests in exponential families.

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