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.
Highlights
In this paper, we establish new results of distribution theory and prove new limit theorems of probability theory
The Poisson-Tweedie mixtures are rigorously introduced by formula (3)
We concentrate on the derivation of local limit theorems, which is customary in the case where one deals with integer-valued r.v.’s, since in view of the jumps of their cumulative distribution functions, the integral limit theorems for such r.v.’s are usually less accurate, which is due to discontinuities to be taken care of
Summary
We establish new results of distribution theory and prove new limit theorems of probability theory. The representations for the probability distributions of Tweedie models in terms of the reduced Wright function φ(ρp, 0; ·) introduced by (5) are given in Vinogradov et al (2012, formulas (3.14) and (3.25)).
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