Parametric quantile regression based on the generalized gamma distribution

We explore a particular fully parametric approach to quantile regression and show that this approach can be very successful. Motivated by the provision of reference charts, we work in the specific context of a positive response variable, whose conditional distribution is modelled by the generalized gamma distribution, and a single covariate, the dependence of parameters of the generalized gamma distribution on which is through simple linear and log-linear forms. With only six parameters at most, such models allow a perhaps surprisingly wide range of distributional shapes that seems adequate for many practical situations. We show that maximum likelihood estimation of the models is computationally quite straightforward, that the estimated quantiles behave well, that use of standard maximum likelihood asymptotics to perform likelihood ratio tests of the number of parameters needed and to give pointwise confidence bands based on the expected information matrix are reliable in this context, and we more tentatively provide a simple goodness-of-fit test of the whole model. Two data analyses, from the health and environmental spheres, are included, along with simulation results. We claim that quite a direct parametric maximum likelihood approach like this—which also obviates the problem of quantile crossing—is adequate for many situations, and there is less need than one might think to resort to more complicated semiparametric and non-parametric approaches to quantile regression in practice