Open Access
January, 1975 Invariant Normal Models
Steen Andersson
Ann. Statist. 3(1): 132-154 (January, 1975). DOI: 10.1214/aos/1176343004


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.


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Steen Andersson. "Invariant Normal Models." Ann. Statist. 3 (1) 132 - 154, January, 1975.


Published: January, 1975
First available in Project Euclid: 12 April 2007

zbMATH: 0373.62029
MathSciNet: MR362703
Digital Object Identifier: 10.1214/aos/1176343004

Primary: 62H05
Secondary: 62H10

Keywords: complex and quaternion Wishart distribution , hypothesis given by symmetries in the variance , maximum likelihood estimation , multivariate statistical analysis , Real

Rights: Copyright © 1975 Institute of Mathematical Statistics

Vol.3 • No. 1 • January, 1975
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