Statistical Properties of a Convoluted Beta-Weibull Distribution

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 , , and , i.e. ~BW( , , , ), and has a beta-Weibull distribution expressed as ~BW( , , , ), what then is the distribution of the convolution of and . That is, the distribution of the random variable = + . 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.