Univariate Distributional Analysis with L-moment Statistics using R

This dissertation is the most complete account, to date, of L-moment statistics in the context of univariate distributional analysis using an open-source programming environment---the R environment. The target audience are engineers/scientists with limited backgrounds in statistics and computer programming but with responsibilities in analyzing highly non-Normal, skewed, or heavy-tailed data. The dissertation is written in continuous narrative and is oriented around the software package "lmomco" previously written by the author but tremendously expanded and refined for the dissertation. The dissertation covers an introduction to R and cites the extensive book-literature on R. The dissertation covers, by a large-scale coupling of source code to typeset mathematics, a myriad of topics including quantile functions, order statistics, product moments, probability-weighted moments (PWMs), censored PWMs, L-moments (censored/trimmed), L-comoments, and numerous probability distributions including the two-parameter Cauchy, Exponential, Normal, Gamma, Gumbel, reverse Gumbel, Kumaraswamy, Rayleigh, and Rice; the three-parameter Generalized Extreme Value, Generalized Logistic, Generalized Normal, Generalized Pareto (GPA), right-censored (RC) GPA, trimmed GPA, Pearson Type III, and Weibull; four- and more parameter distributions including the Kappa, Generalized Lambda (GLD), trimmed GLD, and Wakey; and the method of L-moments and method of PWMs for these distributions. L-moment ratio diagrams are thoroughly described and demonstrated in application. Venerable statistics such as Sen Weighted Mean and Gini Mean Difference are considered as are emergent statistics such as Copulas. Extensive simulation studies are shown via code examples and the results are often depicted in figures; these studies demonstrate the reliability of the examples and lmomco by demonstrating consistency with results with the literature. Topical case studies of regional distributional analysis of hydrometeorologic data are shown to guide readers. Particularly new developments by the author (inclusive of newly developed R code following prior literature results) include censored PWMs and L-moments by censoring fraction, threshold, and indicator; the Cauchy, Kumaraswamy, Rayleigh, Rice, trimmed GPA, and RC-GPA distributions; L-comoments in context of Copulas; and theoretical (non-sample) computation of L-moments. Finally, the dissertation provides more than 240 code examples, more than 510 numbered equations, a thorough topical index, and an index of more than 420 R functions used in the examples.