One-Parameter Exponential Families Generated by Transformation Groups

Abstract

We consider a one-parameter exponential family generated by an arbitrary group of transformations of an abstract sample space. Topological assumptions about the group are not required. It is shown that such a family has densities either of the type of the normal distribution or of the type of the gamma distribution with respect to an invariant measure. This is a generalization of results of Dynkin (1951), Lindley (1958) and Ferguson (1962 and 1963).