From The Classical Beta Distribution To Generalized Beta Distributions

Abstract

This Master's project considers on one hand, the construction of the two parameter classical beta distribution on the domain (0, I) and its generalization and on the other hand, the construction of the extended two parameter beta distribution on the domain (0,00) and its generalization. It provides a comprehensive mathematical treatment of these two parameter beta distributions, their constructions, properties, shapes and special cases. There are various ways of constructing distributions in general. Four ways of constructing the classical beta distribution considered in this work are constructions from, I. The definition ofthe beta function; II. A Poisson process; III. The.transformation ofa ratio oftwo independent gamma variables; and IV. Order statistic. Properties derived include the expressions of the rth moments, first four moments, mode, coefficient of skewness and coefficient of kurtosis. Special cases such as Power, Uniform, Arcsine, Triangular shaped, Parabolic shaped and Wigner Semicircle distributions are given. By the transformation technique a two parameter inverted beta distribution was obtained and its special cases such as Lomax (Pareto II) and Log-logistic (Fisk) distributions were constructed. The work also looked at three methods of generalizing the classical beta distribution i.e. through: I. Transformation technique 2. Generator Approach 3. Use of Special functions The transformation technique resulted into the following: I. Three parameter beta distributions of both !irst and second kinds. In particular, the Libby- Novick and McDonald's distributions. II. Generalized four parameter beta distributions of the first and second kinds. Particular cases are the McDonald four parameter beta distribution of the first kind with its special

cases, the McDonald four parameter beta distribution of the second kind with its special cases and the four parameter generalized beta distribution. Ill. The five parameter generalized beta distribution due to McDonald and Xu (1995). Generalized distributions based on generator approach due to the work of Eugene et al. (2002) are classified into: - Beta generated distributions - Exponentiated generated distributions - Generalized beta generated distributions The generalized beta distributions based on the use of special functions considered includes the Confluent hypergeometric distribution and the Gauss Hypergeometric distribution from the Confluent hypergeometric function and the Gauss Hypergeometric function respectively. Other distributions are based on the Appell function and the Bessel function. Tables showing special cases of beta distributions and their generalizations are given in the appendix.