The Annals of Statistics

Invariant Normal Models

Steen Andersson

Full-text: Open access

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

Permanent link to this document
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
links.jstor.org

Subjects
Primary: 62H05: Characterization and structure theory
Secondary: 62H10: Distribution of statistics

Keywords
Multivariate statistical analysis maximum likelihood estimation real complex and quaternion Wishart distribution hypothesis given by symmetries in the variance

Citation

Andersson, Steen. Invariant Normal Models. Ann. Statist. 3 (1975), no. 1, 132--154. doi:10.1214/aos/1176343004. https://projecteuclid.org/euclid.aos/1176343004


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