Characterizing Exponential Family Distributions by Moment Generating Functions

Abstract

It is shown that if $\\mathbf{T}$ has an unknown exponential family distribution with natural parameter $\\mathbf{\\theta}$, then $\\mathbf{G(\\theta)} = \\mathbf{ET}$ uniquely specifies the moment generating function. The converse is proved, namely, if $\\{\\mathbf{T_\\theta}\\}$ is a family of random variables with moment generating functions of a certain form, then it must be an exponential family. Moreover, several necessary and sufficient conditions are given so that a function can be the mean value function of an exponential family distribution.