Journal
MATHEMATICAL METHODS OF STATISTICS
Volume 26, Issue 3, Pages 159-175Publisher
PLEIADES PUBLISHING INC
DOI: 10.3103/S1066530717030012
Keywords
natural exponential families; exponential dispersion models; variance functions; polynomial variance functions; Poisson-overdispersed distribution
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Funding
- Zimmerman foundation
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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.
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