Extended Poisson–Tweedie: Properties and regression models for count data:

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 and 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 the 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, Polya–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 hurdl...