Necessary and Sufficient Conditions for a Statistical Problem to be Invariant Under a Lie Group

Abstract

Although a great deal has been written concerning the theory of tests, decisions and inference for statistical problems invariant under the action of some group, (see for example [4]-[7], [9], [12]-[14], [16]), no great amount of literature exists concerning the problem of discerning whether or not a given problem is actually invariant under some group. In fact the literature seems to consist of one abstract [8] and one paper [15]. In this paper necessary and sufficient conditions are developed that a statistical problem must satisfy in order that it be invariant, in a precise sense to be defined later, under a fairly general class of transformation groups, Lie transformation groups. It must be added, however, that the sufficient conditions are to some extent tautological. In addition two methods of actually constructing the group, if it can be shown to exist, are given, and the main theorem is illustrated by a variety of examples. One of the examples yields the interesting result that the fiducial distribution of the correlation coefficient derived from a sample from a bivariate normal distribution by R. A. Fisher is not a Bayes' distribution for any prior distribution.