Contributions to statistical distribution theory with applications

Abstract

The whole thesis contains 10 chapters. Chapter 1 is the introductory chapter of my thesis and the main contributions are presented in Chapter 2 through to Chapter 9. Chapter 10 is the conclusion chapter. These chapters are motivated by applications to new and existing problems in finance, healthcare, sports, and telecommunications. In recent years, there has been a surge in applications of generalized hyperbolic distributions in finance. Chapter 2 provides a review of generalized hyperbolic and related distributions, including related programming packages. A real data application is presented which compares some of the distributions reviewed. Chapter 3 and Chapter 4 derive conditions for stochastic, hazard rate, likelihood ratio, reversed hazard rate, increasing convex and mean residual life orderings of Pareto distributed variables and Weibull distributed variables, respectively. A real data application of the conditions is presented in each chapter. Motivated by Lee and Cha [The American Statistician 69 (2015) 221-230], Chapter 5 introduces seven new families of discrete bivariate distributions. We reanalyze the football data in Lee and Cha (2015) and show that some of the newly proposed distributions provide better fits than the two families proposed by Lee and Cha (2015). Chapter 6 derives the distribution of amplitude, its moments and the distribution of phase for thirty-four flexible bivariate distributions. The results in part extend those given in Coluccia [IEEE Communications Letters, 17, 2013, 2364-2367]. Motivated by Schoenecker and Luginbuhl [IEEE Signal Processing Letters, 23, 2016, 644-647], Chapter 7 studies the characteristic function of products of two independent random variables. One follows the standard normal distribution and the other follows one of forty other continuous distributions. In this chapter, we give explicit expressions for the characteristic function of products, and some of the results are verified by simulations. Cossette, Marceau, and Perreault [Insurance: Mathematics and Economics, 64, 2015, 214-224] derived formulas for aggregation and capital allocation based on risks following two bivariate exponential distributions. Chapter 8 derives formulas for aggregation and capital allocation for thirty-three commonly known families of bivariate distributions. This collection of formulas could be a useful reference for financial risk management. Chapter 9 derives expressions for the kth moment of the dependent random sum using copulas. It also extends Mao and Zhao[IMA Journal of Management Mathematics, 25, 2014, 421-433]’s results to the case where the components of the sum are not identically distributed. The practical usefulness of the results in terms of computational time and computational accuracy is demonstrated by simulation.

Date of Award	1 Aug 2018
Original language	English
Awarding Institution	The University of Manchester
Supervisor	Georgi Boshnakov (Supervisor) & Saraleesan Nadarajah (Supervisor)