On the Weibull-Gamma frailty model, its infinite moments, and its connection to generalized log-logistic, logistic, Cauchy, and extreme-value distributions

It is shown that the commonly used Weibull-Gamma frailty model has a finite number of finite moments only and that its marginal distribution generalizes the log-logistic distribution. In some cases there is not even a finite variance, and there are cases without a single finite moment. Upon transformation to the entire real line, generalized logistic and generalized Cauchy distributions are introduced and their connection with the previous ones established, as well as with the extreme-value distribution. Apart from intrinsic and classroom value, the family can be of use when formulating non-informative priors in Bayesian data analysis. Also, gauging the amount of finite moments is important when checking regularity conditions in the Weibull-Gamma model. Our findings are illustrated using data from survival in cancer patients.