The Annals of Mathematical Statistics
- Ann. Math. Statist.
- Volume 38, Number 4 (1967), 1061-1067.
Statistical Models and Invariance
Brillinger  gives necessary and sufficient conditions for a model to be invariant under a Lie group of transformations. The problems that can be handled by his conditions are surveyed, and found effectively to be restricted to one-dimensional problems amendable to Lindley's  method and to problems connected with conflicts between Bayes' and fiducial theory. The problem of finding the general model invariant under a given group is proposed. Brillinger's theorem produces differential equations for the model. A general solution can be obtained by direct methods.
Ann. Math. Statist., Volume 38, Number 4 (1967), 1061-1067.
First available in Project Euclid: 27 April 2007
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Fraser, D. A. S. Statistical Models and Invariance. Ann. Math. Statist. 38 (1967), no. 4, 1061--1067. doi:10.1214/aoms/1177698775. https://projecteuclid.org/euclid.aoms/1177698775