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Sufficiency and Exponential Families for Discrete Sample Spaces

Erling Bernhard Andersen
Journal of the American Statistical Association
Vol. 65, No. 331 (Sep., 1970), pp. 1248-1255
DOI: 10.2307/2284291
Stable URL: http://www.jstor.org/stable/2284291
Page Count: 8
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Sufficiency and Exponential Families for Discrete Sample Spaces
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Abstract

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.

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