We introduce a new class of Poisson-exponential-Tweedie (PET) mixture in the framework of generalized linear models for ultra-overdispersed count data. The mean–variance relationship is of the form \(m+m^{2}+\phi m^{p}\), where \(\phi\) and p are the dispersion and Tweedie power parameters, respectively. The proposed model is equivalent to the exponential-Poisson–Tweedie models arising from geometric sums of Poisson–Tweedie random variables. In this respect, the PET models encompass the geometric versions of Hermite, Neyman Type A, Pólya–Aeppli, negative binomial and Poisson–inverse Gaussian models. The algorithms we shall propose allow to estimate the real power parameter, which works as an automatic distribution selection. Instead of the classical Poisson, zero-shifted geometric is presented as the reference count distribution. Practical properties are incorporated into the PET of new relative indexes of dispersion and zero-inflation phenomena. Simulation studies demonstrate that the proposed model highlights unbiased and consistent estimators for large samples. Illustrative practical applications are analysed on count data sets, in particular, PET models for data without covariates and PET regression models. The PET models are compared to Poisson–Tweedie models showing that parameters of both models are adopted to data.

在超线性分布的计数数据的广义线性模型的框架中，我们引入了一类新的Poisson-指数-Tweedie（PET）混合物。均值-方差关系的形式为\（m + m ^ {2} + \ phi m ^ {p} \），其中\（\ phi \）和p分别是色散和Tweedie功率参数。拟议的模型等效于由Poisson-Tweedie随机变量的几何和产生的指数Poisson-Tweedie模型。在这方面，PET模型涵盖了Hermite，Neyman A型，Pólya–Aeppli，负二项式和Poisson逆高斯模型的几何版本。我们将建议的算法允许估计有功功率参数，该参数可作为自动配电选择。代替经典的泊松，将零偏移的几何图形表示为参考计数分布。实用特性已被纳入PET的新的相对色散和零膨胀现象相对指数中。仿真研究表明，所提出的模型突出了大样本的无偏和一致估计量。在计数数据集上分析了示例性实际应用，尤其是没有协变量的数据的PET模型和PET回归模型。将PET模型与Poisson–Tweedie模型进行比较，表明这两个模型的参数都已用于数据。