On Poisson–Tweedie mixtures

Vladimir Vinogradov, Richard B. Paris

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Abstract

Poisson-Tweedie mixtures are the Poisson mixtures for which the mixing measure is generated by those members of the family of Tweedie distributions whose support is non-negative. This class of non-negative integer-valued distributions is comprised of Neyman type A, back-shifted negative binomial, compound Poisson-negative binomial, discrete stable and exponentially tilted discrete stable laws. For a specific value of the “power” parameter associated with the corresponding Tweedie distributions, such mixtures comprise an additive exponential dispersion model. We derive closed-form expressions for the related variance functions in terms of the exponential tilting invariants and particular special functions. We compare specific Poisson-Tweedie models with the corresponding Hinde-Demétrio exponential dispersion models which possess a comparable unit variance function. We construct numerous local approximations for specific subclasses of Poisson-Tweedie mixtures and identify Lévy measure for all the members of this three-parameter family.
Original languageEnglish
Number of pages23
JournalJournal of Statistical Distributions and Applications
Volume4
Issue number14
Early online date2 Oct 2017
DOIs
Publication statusPublished - Dec 2017

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