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May, 1975 Characterizing Exponential Family Distributions by Moment Generating Functions
Allan R. Sampson
Ann. Statist. 3(3): 747-753 (May, 1975). DOI: 10.1214/aos/1176343140

Abstract

It is shown that if $\mathbf{T}$ has an unknown exponential family distribution with natural parameter $\mathbf{\theta}$, then $\mathbf{G(\theta)} = \mathbf{ET}$ uniquely specifies the moment generating function. The converse is proved, namely, if $\{\mathbf{T_\theta}\}$ is a family of random variables with moment generating functions of a certain form, then it must be an exponential family. Moreover, several necessary and sufficient conditions are given so that a function can be the mean value function of an exponential family distribution.

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Allan R. Sampson. "Characterizing Exponential Family Distributions by Moment Generating Functions." Ann. Statist. 3 (3) 747 - 753, May, 1975. https://doi.org/10.1214/aos/1176343140

Information

Published: May, 1975
First available in Project Euclid: 12 April 2007

zbMATH: 0303.62010
MathSciNet: MR373094
Digital Object Identifier: 10.1214/aos/1176343140

Subjects:
Primary: 62E10
Secondary: 62H05

Keywords: characterization of distributions , exponential , exponential family , moment generating function , normal , Wishart

Rights: Copyright © 1975 Institute of Mathematical Statistics

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Vol.3 • No. 3 • May, 1975
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