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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关于泊松-特威迪混合物

Poisson-Tweedie混合物是泊松混合物，其混合度量是由Tweedie分布族的支持非负的那些成员生成的。此类非负整数值分布包括Neyman A型，后移负二项式，复合Poisson负二项式，离散稳定和指数倾斜离散稳定定律。对于与相应的Tweedie分布关联的“功率”参数的特定值，此类混合物包括加法指数色散模型。我们根据指数不变量和特定特殊函数推导了相关方差函数的闭式表达式。我们将特定的Poisson-Tweedie模型与相应的Hinde-Demétrio指数色散模型进行比较，后者具有可比的单位方差函数。我们为Poisson-Tweedie混合物的特定子类构造了许多局部逼近，并为该三参数族的所有成员确定了Lévy度量。