## The Annals of Statistics

### 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.

#### Article information

Source
Ann. Statist., Volume 3, Number 1 (1975), 132-154.

Dates
First available in Project Euclid: 12 April 2007

https://projecteuclid.org/euclid.aos/1176343004

Digital Object Identifier
doi:10.1214/aos/1176343004

Mathematical Reviews number (MathSciNet)
MR362703

Zentralblatt MATH identifier
0373.62029

JSTOR