Application of Three Parameter Generalized Beta I Distribution in Binomial Mixture

Abstract

Two or more individual distributions can be mixed together to form a new distribution. According to Feller, this can be done using weights that sum up to unit. Also by considering a parameter which is a random variable taking another distribution then a new distribution ca be formed. The nature of the mixing distribution has effect on the new distribution formed. If the mixing distribution is continuous random variable, then the new mixture formed is also continuous. Skellam pioneered this study when he constructed binomial mixture with varying parameter also taking beta distribution. McDonald generalized beta distribution to a three parameter. This paper focusses on binomial mixture with a three parameter generalized beta distribution considered as the mixing distribution. The three parameters considered are McDonald and Libby-Novick three parameter beta distributions. Two methods of construction of binomial mixture are discussed and proved to obtain identical results. The distribution derived is proved to be a probability density function. Moments of the mixture are obtained. The binomial mixture obtained is useful in addressing the challenges of over-dispersion which is common with probability distributions having binomial outcome.