Optimale vorhersage und schätznng in regulären und singulären regressionsmodellen

Abstract

We consider arbit,rary regression models y = Xβ + e, where the covariance matrix from e can also be singulary. First we determine optimal homogeneous and inhomogeneous linear predictions for y ∗where the true equation for y ∗ be y ∗ = Xβ + e ∗ As the risk we use a mean quadratic (not necessary) distance. Then we study the possibilities for improving the best homogeneous unbiased prediction. We show, that this is possible, if there are additional informations about X β if for instance ∣X β∣ is bounded. After this me consider the improvement of a homogeneous prediction by admitting inhomogeneous predictions. In a further section we determine for (may be even singulary) models y = X β + e a GAUSS-MARKOV estimation for β and we get also statements of the question in which cases this estimation can be improved. Thereby the strict connection between unbiased predictions and G.-M. estimations is used.