Double Generalized Linear Models

Double generalized linear regression models (DGLMs) are defined in the two-parameter exponential family of distributions, obtained by assuming unknown dispersion parameters in the one-parameter exponential family of distributions. Some examples of DGLMs considered here are the heteroscedastic normal regression models, mean and variance gamma regression models, and mean and dispersion beta regression models, where the mean and dispersion parameters follow regression structures. The models are applied to analyze (i) the volume of useful wood of cherry trees, with the height and diameter of each tree as covariates, using normal heteroscedastic regression models; (ii) property tax data, assuming a gamma regression, where the mean and shape (or variance) parameters follow regression structures; and (iii) ratios and proportions, where the mean and dispersion parameters follow regression structures, using beta regression models. The maximum likelihood and Bayesian methods used to obtain parameter inferences are presented and used in various applications.