Submission note: A thesis submitted in total fulfilment of the requirements for the degree of Doctor of Philosophy [to the] Department of Mathematics and Statistics, School of Engineering and Mathematical Sciences, College of Science, Health and Engineering, La Trobe University, Victoria, Australia.
Statistical modeling and estimation of quantiles is an integral part of statistical data analysis in interpreting real-world phenomena. The purpose of this thesis is to contribute to the existing body of knowledge in quantile-based methods in modeling and estimation while providing simulation studies and real data applications supporting the new contributions. The results are discussed coherently in the thesis, including original publications that have been either published, accepted to be published or submitted for peer review. In the first part of the thesis, we propose a new approach based on the Probability Density Quantile (pdQ) for parameter estimation of the Generalized Lambda Distribution (GLD). Defined in terms of a location parameter, scale parameter, and two shape parameters, the GLD is widely used for modeling in many fields because of its flexibility in being able to mimic many other distributions. However, due to there being four parameters, choosing optimal parameters and/or estimating those parameters is not straightforward. We compare our pdQ approach with the existing methods in regards to time efficiency and performance. Further, we extend the introduced method for the Generalized Beta Distribution, illustrating the applicability of the method more broadly than just the GLD. In the second part of the thesis, we introduce several methods, including a method based on the GLD, to obtain confidence intervals for quantiles when only a frequency distribution or histogram is available. These methods are extended to measuring inequality for grouped income data where data is often provided in such summary format to protect the confidentiality of individuals. Here we show that interval estimators for quantile-based inequality measures are suited to this type of data. The thesis also includes two web-based Shiny Applications for end-users to apply these methods in their research.
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