Sufficient Statistics and Discrete Exponential Families

Abstract

$\{P_\theta\}$ is a set of probabilities on a countable set $_\chi$ such that $P_\theta(x) > 0$ for each $x$ and $\theta$. We prove that if $\{P_\theta\}$ is not an exponential family, then each sufficient statistic for $n$ independent observations must be one-to-one, modulo permutations, on an infinite product set (which does not depend on the sufficient statistic).