Normal distribution assumption and least squares estimation function in the model of polynomial regression

Kurzfassung/Abstract

In a linear model $Y=X\beta +Z$ a linear functional $\beta \mapsto \gamma '\beta$ is to be estimated under squared error loss. It is well-known that, provided Y is normally distributed, the ordinary least squares estimation function minimizes the risk uniformly in the class ${\cal P}$ of all equivariant estimation functions and is admissible in the class ${\cal E}$ of all unbiased estimation functions. For the design matrix X of a polynomial regression set up it is shown for almost all estimation problems that the ordinary least squares estimation function is uniformly best in ${\cal P}$ and also admissible in ${\cal E}$ only if Y is normally distributed.