ON THE CHOICE OF THE LOSS FUNCTION IN COVARIANCE ESTIMATION

In a generalized linear model, under certain conditions} the covariante matrices of a two-stage Aitken estimator and the Gauss-Markov estimator are related via Kariya's inequality of the form Cov(l(ü)) á Cov(B(ü)) s ε [C(y,y) ]Cov($(Ω)), liiere the true covarianoe matrix Ω of the response is a function of an estimable parameter y. This inequality is used as a basis for defining a loss function to estimate y. Two models are analyzed: The seemingly unrelated regressions and the heteroscedaetic model. In both cases the loss function is invariant with respect to an appropriate group of transformations and the minimum risk equivariant estimator is obtained. This approach reduces the degree of arbitrariness for choosing a loss function. AMS 1980 subject classifications: 62 Η 11, 62 C 05