The product method for finding bivariate Kummer beta distributions

Abstract

This study systematically derives bivariate Kummer beta distributions for the first time using the product method. Although the definition of the bivariate product distribution is stated in general as the product of any two functions, f(.) and h(.), this study looks at a special case where f(.) and h(.), are taken to be kernels of known distributions. Particularly, this study considers the case where f(x_1,x_2) is taken to be kernels from various bivariate beta distributions and h(x_1,x_2) is taken to be the product of two exponential kernels, i.e. h(x_1,x_2)=e^(-øx_1) e^(-øx_2)=e^(-ø(x_1+x_2)). The bivariate beta distributions that are considered include: the bivariate beta type I, bivariate generalized beta type I, bivariate beta type III, bivariate beta type IV, bivariate extended beta type IV and the bivariate beta type V distribution. The new bivariate product distributions that are constructed in this way are referred to as bivariate Kummer beta distributions. The word Kummer originates from the fact that the normalizing constant, K, of the pdf's contain the Kummer function (also referred to as the confluent hypergeometric function) or a related form of it. These new bivariate Kummer beta distributions have the original bivariate beta distribution parameters as well as the parameter, ø. When ø is set equal to 0, the bivariate Kummer beta type distributions simplify to the bivariate beta distribution whose kernel was used in its construction. This study derives the joint, marginal and conditional pdf's of these distributions. The effect of the parameter, ø, on the correlation between X_1 and X_2, the joint pdf and the marginal pdf is investigated graphically. Finally, two examples of possible applications are provided.