# Parametric quantile regression based on the generalised gamma distribution

Noufaily, Angela (2011). Parametric quantile regression based on the generalised gamma distribution. PhD thesis The Open University.

## Abstract

Quantile regression offers an extension to regression analysis where a modified version of the least squares method allows the fitting of quantiles at every percentile of the data rather than the mean only. Using the well-known three-parameter generalised gamma distribution to model variation in data, we present a parametric quantile regression study for positive univariate reference charts. The study constitutes an overall package that includes all different stages of parametric modeling starting from model identification to parameter estimation, model selection and finally model checking.

We improve on earlier work by being the first to formulate the iterative approach to solution of the likelihood score equations of the generalised gamma distribution in such a way that the individual equations involved are uniquely solvable and far from being problematic as a number of authors have suggested. We conduct likelihood ratio tests to choose the best model within the three-, four-, five- and six-parameter generalised gamma family obtained by making the parameters linearly (or loglinearly) dependent on a univariate covariate. Quantiles are plotted accordingly and asymptotic theory for obtaining the expressions for confidence bands around them is given. Based on the chi-square goodness-of-fit test, we suggest a test statistic that checks the goodness of the generalised gamma model for given data. We validate the whole theoretical process computationally via simulations. Lastly, we demonstrate the different steps of the proposed modeling procedure through two main applications; one is environment-related and the other health-related.

In a parallel fashion, inspired by the generalised gamma distribution, we introduce an alternative three-parameter distribution with useful statistical properties. We explore briefly maximum likelihood estimation and asymptotic theory of the alternative distribution and we compare it computationally to the generalised gamma.

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