A surprising property of uniformly best linear affine estimation in linear regression when prior information is fuzzy

It is already shown in Arnold and Stahlecker (2009) that, in linear regression, a uniformly best estimator exists in the class of all Γ-compatible linear affine estimators. Here, prior information is given by a fuzzy set Γ defined by its ellipsoidal α-cuts. Surprisingly, such a uniformly best linear affine estimator is uniformly best not only in the class of all Γ-compatible linear affine estimators but also in the class of all estimators satisfying a very weak and sensible condition. This property of a uniformly best linear affine estimator is shown in the present paper. Furthermore, two illustrative special cases are mentioned, where a generalized least squares estimator on the one hand and a general ridge or Kuks-Olman estimator on the other hand turn out to be uniformly best.