#### Date of Award

2011

#### Degree Name

Mathematics

#### College

College of Science

#### Type of Degree

M.A.

#### Document Type

Thesis

#### First Advisor

Alfred Akinsete

#### Second Advisor

Ari Aluthge

#### Third Advisor

Basant Karna

#### Abstract

A new class of distributions recently developed involves the logit of the beta distribution. Among this class of distributions are the beta-normal (Eugene et.al. (2002)); beta-Gumbel (Nadarajah and Kotz (2004)); beta-exponential (Nadarajah and Kotz (2006)); beta-Weibull (Famoye et al. (2005)); beta-Rayleigh (Akinsete and Lowe (2008)); beta-Laplace (Kozubowski and Nadarajah (2008)); and beta-Pareto (Akinsete et al. (2008)), among a few others. Many useful statistical properties arising from these distributions and their applications to real life data have been discussed in the literature. One approach by which a new statistical distribution is generated is by the transformation of random variables having known distribution function(s). The focus of this work is to investigate the statistical properties of the convoluted beta-Weibull distribution, defined and extensively studied by Famoye et al. (2005). That is, if X is a random variable having the beta-Weibull distribution with parameters a1, B1, c1, y1 i.e. X=BW(a1, B1, c1 and y1) and Y has a beta-Weibull distribution expressed as Y=BW(a2, B2, c2, y2) what then is the distribution of the convolution of X and Y. That is, the distribution of the random variable Z=X+Y. We obtain the probability density function (pdf) and the cumulative distribution function (cdf) of the convoluted distribution. Various statistical properties of this distribution are obtained, including, for example, moment, moment and characteristic generating functions, hazard function, and the entropy. We propose the method of Maximum Likelihood Estimation (MLE) for estimating the parameters of the distribution. The open-source software R is used extensively in implementing our results.

#### Subject(s)

Mathematical statistics.

#### Recommended Citation

Sun, Jianan, "Statistical Properties of a Convoluted Beta-Weibull Distribution" (2011). *Theses, Dissertations and Capstones*. 277.

https://mds.marshall.edu/etd/277

#### Included in

Analysis Commons, Logic and Foundations Commons, Number Theory Commons, Statistics and Probability Commons