On Poisson–Tweedie mixtures

Vladimir V. Vinogradov () and Richard B. Paris ()
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Vladimir V. Vinogradov: Ohio University
Richard B. Paris: Abertay University

Journal of Statistical Distributions and Applications, 2017, vol. 4, issue 1, 1-23

Abstract: 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.

Keywords: Discrete stable distribution; Invariant of the exponential tilting transformation; Lambert W function; Large deviations; Lévy measure; Natural exponential family; Poisson-Tweedie mixture; Refined local limit theorem; Variance function; Wright function; 33E20; 60E05; 60E07; 60F05; 60F10 (search for similar items in EconPapers)
Date: 2017
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DOI: 10.1186/s40488-017-0068-1

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