Invariant Normal Models

Abstract

Many hypotheses in the multidimensional normal distribution are given or can be given by symmetries or, in other words, invariance. This means that the variances are invariant under a given subgroup of the general linear group in the vector space of observations. In this paper we define a class of hypotheses, the Invariant Normal Models, including all symmetry hypotheses. We derive the maximum likelihood estimator of the mean and variance and its distribution under the hypothesis. The value of the paper lies in the mathematical formulation of the theory and in the general results about hypotheses given by symmetries. Especially the formulation gives an easy simultaneous derivation of the real, complex and quaternion version of the Wishart distribution. Furthermore, we show that every invariant normal model with mean-value zero can be obtained by a symmetry.