Parametric Deviations in Linear Models

The classical result about the optimality of the OLS (ordinary least squares) estimator $$\hat \beta $$ is that it is BLUE (best linear unbiased estimator) in a general linear model N(Xβ,V) when there exists a matrix B with VX=XB. The covariance-matrix of $$\hat \beta $$ of course depends on V in that case and generally is not equal to the covariance matrix of $$\hat \beta $$ in the classical linear model N(Xβ,σ2l). It is shown that this is the case iff VX=αX (i.e. the column space of X is not only invariant under V but is contained in some eigenspace of V). Furthermore a characterization of normal linear models for which the usual F-test is the UMP-test invariant under a certain class of orthogonal transformations (as it is for the classical linear model) is given in terms of eigenvalues of the covariance-matrix. Some inequalities connecting the F-statistic of the classical and the general linear model based on some matrix-norms of V-αl are derived and the results are applied to analysis of variance problems with correlated observations.