Sufficiency and Exponential Families for Discrete Sample Spaces

All known theorems of the Darmois-Koopman-Pitman type are based on the assumption of absolutely continuous probabilities. In this article, a Darmois-Koopman-Pitman type theorem is proved for families of probability measures that are defined on discrete sample spaces. Two assumptions are imposed on the sufficient statistic for the family: (1) the range space of the sufficient statistic can be completely ordered, and (2) this ordering of the range space of the sufficient statistic has a certain regularity property. Under (1) and (2), it can be proved that the given family is exponential of order 1.