A characterization of the normal distribution by sufficiency of the least squares estimation

Kurzfassung/Abstract

It is proved that if $y=\theta +Z$, where the unknown parameter $\theta$ is an element of a linear subspace V of $R\sp n$ and Z is a nondegenerate random vector with independent components each having mean zero, then, under fairly general conditions on V, the projection of $R\sp n$ on V (with respect to a certain scalar product) is a sufficient statistic for the class of distributions of y if and only if Z is normally distributed. \par The condition of independence of components of Z can not be relaxed but the authors succeed in extending the above result to the set-up of the multivariate linear model.