On the mean value parametrization of natural exponential families — a revisited review

S. K. Bar-Lev; C. C. Kokonendji

doi:10.3103/S1066530717030012

On the mean value parametrization of natural exponential families — a revisited review

S. K. Bar-Lev, C. C. Kokonendji

School of Political Sciences

Research output: Contribution to journal › Review article › peer-review

Abstract

It is well known that any natural exponential family (NEF) is characterized by its variance function on its mean domain, often much simpler than the corresponding generating probability measures. The mean value parametrization appeared to be crucial in some statistical theory, like in generalized linear models, exponential dispersion models and Bayesian framework. The main aim of the paper is to expose the mean value parametrization for possible statistical applications. The paper presents an overview of the mean value parametrization and of the characterization property of the variance function for NEF’s. In particular it introduces the relationships existing between the NEF’s generating measure, Laplace transform and variance function as well as some supplemental results concerning the mean value representation. Some classes of polynomial variance functions are revisited for illustration. The corresponding NEF’s of such classes are generated by counting probabilities on the nonnegative integers and provide Poisson-overdispersed competitors to the homogeneous Poisson distribution.

Original language	English
Pages (from-to)	159-175
Number of pages	17
Journal	Mathematical Methods of Statistics
Volume	26
Issue number	3
DOIs	https://doi.org/10.3103/S1066530717030012
State	Published - 1 Jul 2017

Bibliographical note

Funding Information:
We thank a referee for helpful comments which improved the presentation of the paper. Part of this work was done while the second author was a visitor at the University of Haifa and supported by the Zimmerman foundation. The paper was completed while S. K. Bar-Lev was a visiting professor at NYU Shanghai, China, during the Spring semester of 2017. We sincerely thank Gérard Letac for illuminating and interesting discussion and comments.

Publisher Copyright:
© 2017, Allerton Press, Inc.

Keywords

Poisson-overdispersed distribution
exponential dispersion models
natural exponential families
polynomial variance functions
variance functions

ASJC Scopus subject areas

Statistics and Probability
Statistics, Probability and Uncertainty

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10.3103/S1066530717030012

Cite this

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abstract = "It is well known that any natural exponential family (NEF) is characterized by its variance function on its mean domain, often much simpler than the corresponding generating probability measures. The mean value parametrization appeared to be crucial in some statistical theory, like in generalized linear models, exponential dispersion models and Bayesian framework. The main aim of the paper is to expose the mean value parametrization for possible statistical applications. The paper presents an overview of the mean value parametrization and of the characterization property of the variance function for NEF{\textquoteright}s. In particular it introduces the relationships existing between the NEF{\textquoteright}s generating measure, Laplace transform and variance function as well as some supplemental results concerning the mean value representation. Some classes of polynomial variance functions are revisited for illustration. The corresponding NEF{\textquoteright}s of such classes are generated by counting probabilities on the nonnegative integers and provide Poisson-overdispersed competitors to the homogeneous Poisson distribution.",

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On the mean value parametrization of natural exponential families — a revisited review

Abstract

Bibliographical note

Keywords

ASJC Scopus subject areas

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