Improved estimation and regression techniques with the generalised lambda distribution

Generalised distributions are appealing because of their flexibility (compared to standard, non-generalised distributions). The generalised lambda distribution (gλd) is a particularly appealing generalised distribution because i) it provides a wide range of shapes in one distributional form, ii) it has quality software available for its use, and iii) it has extensive literature on its estimation. This thesis focuses on the so-called FKML gλd (Freimer et al., 1988) as it has a number of advantages over alternative parameterisations of the gλd. Two important areas of gλd research are enhanced by this thesis. The first area is estimation of the gλd, which is an important area as it is fundamental to using the gλd. The second area is gλd-based regression modelling, which is an important area as it can be more informative and more accurate than techniques which treat the error term nonparametrically. The need to further explore parametric regression modelling was expressed by Gilchrist (2008): "the statistical community has some unfinished business in the parametric modelling of the complete regression model". This thesis advances estimation of the gλd in four ways. Firstly, it develops a new algorithm for the starship method which overcomes its slow speeds. Secondly, it develops a new algorithm for numerical maximum likelihood that improves its fitting of heavy-tailed data. Thirdly, it overcomes technical difficulties of numerical maximum likelihood by implementing two alternative methods with desirable properties: maximum spacings product and Titterington's method. Fourthly, it develops the first algorithm for the method of trimmed L-moments for the FKML gλd. This thesis advances gλd-based regression modelling by developing three new techniques where the error term follows the FKML gλd. Firstly, Versatile Regression improves King et al.'s (2005) "starship regression" by overcoming its slow speeds and accommodating a range of estimation methods. Secondly, Quantile Versatile Regression extends Versatile Regression to the case where any conditional quantile can be fitted and produces, in many cases, more accurate prediction intervals than quantile regression (Koenker and Bassett, 1978). Thirdly, Stretched Regression fits data that is known to have a lower bound on the response variable, and illustrates how the FKML gλd can be used to develop tailored regression models.