Characterization and classification of multiple stable Tweedie models

Extending normal stable Tweedie models, the multiple stable Tweedie (MST) models are recently introduced as a huge class of multivariate distributions. They are composed by a fixed univariate stable Tweedie variable having a positive mean domain and random variables that, given the fixed one, are real independent stable Tweedie variables, possibly different, with the same dispersion parameter equal to the fixed component.Within the framework of exponential dispersion models, we completely prove the characterization of the MST models through their variance functions under steepness property. Thereforewe deduce a new classification of the Poisson-MST, gamma-MST, noncentral-gamma-MST, and inverse-Gaussian-MST families, where each of them contains one element of the normal stable Tweedie models, namely normal-Poisson, normal-gamma (or gamma-Gaussian), normal-noncentral-gamma, and normal-inverse-Gaussian distributions, respectively.