# Extended Poisson-Tweedie: properties and regression models for count data

arXiv: https://arxiv.org/abs/1608.06888 · HTML: https://mlsanity.com/paper/1608.06888/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 24 Aug 2016.

## Built on by (most cited first, 3 of 3)

- [Statistical Modelling of Level Difficulty in Puzzle Games](https://mlsanity.com/paper/2107.03305/) (2021)
- [Distribution-free inference for LightGBM and GLM with Tweedie loss](https://mlsanity.com/paper/2507.06921/) (2025)
- [Predicting integers from continuous parameters](https://mlsanity.com/paper/2602.10751/) (2026)

## Abstract (arXiv)

We propose a new class of discrete generalized linear models based on the class of Poisson-Tweedie factorial dispersion models with variance of the form μ + φμp, where μ is the mean, φ and p are the dispersion and Tweedie power parameters, respectively. The models are fitted by using an estimating function approach obtained by combining the quasi-score and Pearson estimating functions for estimation of the regression and dispersion parameters, respectively. This provides a flexible and efficient regression methodology for a comprehensive family of count models including Hermite, Neyman Type A, P'olya-Aeppli, negative binomial and Poisson-inverse Gaussian. The estimating function approach allows us to extend the Poisson-Tweedie distributions to deal with underdispersed count data by allowing negative values for the dispersion parameter φ. Furthermore, the Poisson-Tweedie family can automatically adapt to highly skewed count data with excessive zeros, without the need to introduce zero-inflated or hurdle components, by the simple estimation of the power parameter. Thus, the proposed models offer a unified framework to deal with under, equi, overdispersed, zero-inflated and heavy-tailed count data. The computational implementation of the proposed models is fast, relying only on a simple Newton scoring algorithm. Simulation studies showed that the estimating function approach provides unbiased and consistent estimators for both regression and dispersion parameters. We highlight the ability of the Poisson-Tweedie distributions to deal with count data through a consideration of dispersion, zero-inflated and heavy tail indices, and illustrate its application with four data analyses. We provide an textttR implementation and the data sets as supplementary materials.