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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Poisson Mixture
Exponential Dispersion Model
Negative Binomial
Variance Function
Exponential Tilting
Non-negative
Compound Poisson
Mixture Distribution
Stable Laws
Local Approximation
Poisson Model
Special Functions
Closed-form
Unit
Integer
Invariant
Family

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Vinogradov, Vladimir ; Paris, Richard B. / On Poisson–Tweedie mixtures. In: Journal of Statistical Distributions and Applications. 2017 ; Vol. 4, No. 14.
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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-Dem{\'e}trio exponential dispersion models which possess a comparable unit variance function. We construct numerous local approximations for specific subclasses of Poisson-Tweedie mixtures and identify L{\'e}vy measure for all the members of this three-parameter family.",
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Vinogradov, V & Paris, RB 2017, 'On Poisson–Tweedie mixtures', Journal of Statistical Distributions and Applications, vol. 4, no. 14. https://doi.org/10.1186/s40488-017-0068-1

On Poisson–Tweedie mixtures. / Vinogradov, Vladimir; Paris, Richard B.

In: Journal of Statistical Distributions and Applications, Vol. 4, No. 14, 12.2017.

Research output: Contribution to journal › Article

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AU - Paris, Richard B.

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N2 - 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.

AB - 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.

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Vinogradov V, Paris RB. On Poisson–Tweedie mixtures. Journal of Statistical Distributions and Applications. 2017 Dec;4(14). https://doi.org/10.1186/s40488-017-0068-1

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