Asymmetric generalizations of symmetric univariate probability distributions obtained through quantile splicing

Abstract

This thesis develops a skewing methodology for the formulation of two-piece families of distri- butions that can be defined through their cumulative distribution functions (CDFs), probability density functions (PDFs) or quantile functions. The advantage of this methodology is that the families of distributions constructed have skewness-invariant measures of kurtosis, allowing for the independent analysis of the skewness and kurtosis of a distribution. The central contribution of this thesis is in the development of the quantile function of the two-piece family of distributions. This quantile function is constructed through the use of the quantile functions of half distributions developed from symmetric univariate distributions (henceforth referred to as the parent distribution). This quantile function is the used to derive a general formula for the rth order L-moments of the two-piece family of distributions. The results of these L-moments will be in terms of the L-moments of both the parent distribution and the half distribution. The parameters of this new family of distributions can be estimated through the method of L-moments since closed form expressions exist for the L-moments and subsequently the estimators. The results from the skewing methodology as well as from the formula for the rth order L-moments will be applied to well-known symmetric univariate distributions. These include the arcsine, uniform, cosine, normal, logistic, hyperbolic secant and Student’s t(2) distributions, which do not have a shape parameter, as well as the quantile-based Tukey lamba distribution which has a kurtosis parameter.