Characterizing Exponential Family Distributions by Moment Generating Functions.

摘要

It is shown that if T has an unknown exponential family distribution with natural parameter theta, then G(theta)=ET uniquely specifies the moment generating function. The converse is proved, namely, if T sub 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.