CONTRIBUTION TO DISTRIBUTION THEORY WITH APPLICATIONS

Abstract

The whole thesis contains 9 chapters. Chapter 1 is the introductory chapter of my thesis and the main contributions are in Chapter 2 through to Chapter 9. The theme of these chapters is developing and reviewing statistical distributions. A comprehensive review of transmuted distributions is provided in Chapter 2. Nearly thirty such distributions are reviewed. Real data applications are provided comparing the reviewed distributions to other classes of distributions. This review could serve as an important reference and encourage developments of further transmuted distributions that could model complicated phenomena more accurately. Mukhopadhyay and Cicconetti (2004) derived the Maximum Likelihood Estimator (MLE) and the Uniformly Minimum Variance Unbiased Estimator (UMVUE) of $\theta$ in $N (\theta, \theta)$ and discussed their application to purely sequential and two-stage bounded risk estimation of $\theta$. In Chapter \ref{sachap}, a much simpler expression is derived for the UMVUE of $\theta$. Using this expression, a comprehensive investigation is provided for comparing the performances of the sequential estimators based on the MLE and the UMVUE. Ferreira, Dionisio \& Correia [Physica A: Statistical Mechanics and Its Applications, 505, 2018, 680-687] showed that African stock markets at different time frames (before the Lehman Brothers financial crisis, during the crisis, and after the crisis) do not satisfy the efficient market hypothesis. Chapter 4 provides evidence by means of six different nonparametric tests, and the fit of GARCH(1, 1), TGARCH(1, 1) and EGARCH(1, 1) models accounting for day of the week and month of the year effects that the majority of African stock markets do comply with the efficient market hypothesis. Motivated by Huang [Physica A, 482, 2017, 173-180], Chapter 5 analyses impact factor data in all subject categories for each year from 2010 to 2015. The results are provided by subject categories and by years. The two exponent distribution due to Mansilla et al. [Journal of Informetrics, 1, 2007, 155-160] and the normal distribution are shown to give the best fits for the data sets. The best fits are assessed in terms of probability plots, quantile plots and five other criteria. Dombi, J\'{o}n\'{a}s, T\'{o}th and \'{A}rva [Quality and Reliability Engineering International, 2018, doi: 10.1002/qre.2425] introduced the omega probability distribution. They stated that main properties of this distribution are ``difficult to deal with'' analytically. Chapter 6 derives closed form expressions for all of the main properties of the omega distribution. A real data illustration of the derived expressions is also given. Based on an alpha power transformation method developed in Mahdavi and Kundu [Communications in Statistics---Theory and Methods, 46, 2017, doi: 10.1080/03610926.2015.1130839], Dey, Alzaatreh, Zhang and Kumar [Ozone: Science and Engineering, 39, 2017, 273-285] introduced a novel three-parameter distribution, studied its properties including estimation issues and illustrated an application to an ozone data set. Chapter 7 derives closed form expressions for moment properties of the distribution. The chapter also revisits their data application. Lo [The American Statistician, 2018, doi: 10.1080/00031305.2018.1497541] derived the mean of any integrable random variable in terms of tail probabilities. Chapter 8 extends Lo's formulae for moments of general order in the univariate and bivariate cases. The chapter also derives a multivariate version of Lo's formulae. Examples are given to illustrate the formulae. General relations expressing factorial moments in terms of raw moments, raw moments in terms of factorial moments, factorial moments in terms of central moments and central moments in terms of factorial moments are derived in Chapter 9. Illustrations are given.

Date of Award	1 Aug 2020
Original language	English
Awarding Institution	The University of Manchester
Supervisor	Georgi Boshnakov (Supervisor) & Saraleesan Nadarajah (Supervisor)