Compound gamma bivariate distributions

Summary

Bivariate distributions, which may be of special relevance to the lifetimes of two components of a system, are derived using the following approach. As the two components are part of one system and therefore exposed to similar conditions of service, there will be similarity between their lifetimes that is not shared by components belonging to different systems. The lifetime distribution for a given system is assumed to be Gamma in form (this includes the exponential as a special case; extension to the Stacey distribution, which includes the Weibull distribution, is straightforward). The scale parameter of this distribution is itself a random variable, with a Gamma distribution. We thus obtain what might be termed a compound Gamma-Gamma bivariate distribution. The cumulative distribution function of this may be expressed in terms of one of the double hypergeometric functions of Appell.

Generalised hypergeometric functions play an important part in this paper, and one of Saran's triple hypergeometric functions is obtained when generalising the above model to permit the scale parameters of the distributions for the two components to be correlated, rather than identical.