Note on a Theorem of Dynkin on the Dimension of Sufficient Statistics

Abstract

We show that the existence of a continuous minimal sufficient statistic not equivalent to the order statistics, for $n \geqq 2$ independent observations, is not a sufficient condition for the family of densities, assumed to be Lipschitz, to be an exponential family. This result is intended to be compared with a theorem of Dynkin (p. 24 of [3]) which asserts that the existence of a sufficient statistic not equivalent to the order statistics implies that the family of densities is an exponential family, provided that the densities possess continuous derivatives.