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

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

Research output: Contribution to journalReview articlepeer-review


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 languageEnglish
Pages (from-to)159-175
Number of pages17
JournalMathematical Methods of Statistics
Issue number3
StatePublished - 1 Jul 2017

Bibliographical note

Publisher Copyright:
© 2017, Allerton Press, Inc.


  • 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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