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Journal Article

# Sufficient Statistics and Discrete Exponential Families

J. L. Denny
The Annals of Mathematical Statistics
Vol. 43, No. 4 (Aug., 1972), pp. 1320-1322
Stable URL: http://www.jstor.org/stable/2239961
Page Count: 3

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{Pθ} is a set of probabilities on a countable set χ such that $P_\theta(x) > 0$ for each x and θ. We prove that if {Pθ} 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).