## 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 |
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Pages (from-to) | 159-175 |

Number of pages | 17 |

Journal | Mathematical Methods of Statistics |

Volume | 26 |

Issue number | 3 |

DOIs | |

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