Overdispersed and underdispersed Poisson generalizations

We consider a wide set of statistical models that extend the Poisson distribution. These models are obtained through weighted versions of the Poisson family and can be approximated by a log-linear model. Under general conditions, we prove that the new models contain overdispersed and underdispersed distributions and that they can be parametrized with the mean and variance. A classical data set is analyzed to show the usefulness of the new models.