On a Theorem of Morimoto Concerning Sufficiency for Discrete Distributions

Abstract

We prove in a discrete setting that if for all test functions, $t$, there is a $\mathbf{B}$ measurable test function, $s$, such that $E_p(t) = E_p(s)$ for all $p \in P$ then some subfield of $\mathbf{B}$ is sufficient for $P$.