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.
Citation
J. L. Denny. "Note on a Theorem of Dynkin on the Dimension of Sufficient Statistics." Ann. Math. Statist. 40 (4) 1474 - 1476, August, 1969. https://doi.org/10.1214/aoms/1177697518
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