Bivariate Exponential Distributions

Abstract

This chapter, for simplicity, considers only a number of bivariate exponential (BVE) distributions. Many of the models have multivariate extensions. However, the notations become quite complex. E. J. Gumbel is one of the first to consider bivariate exponential distributions. He proposed three, which are absolutely continuous models. J. Freund is one of the first to consider a model that is physically motivated. The model proposed by H. W. Block and A. P. Basu is closely related to the BVE and the model proposed by Freund. The BVE has univariate exponential marginals, and it satisfies the bivariate loss of memory property (BLMP). Block and Basu showed that it is not possible to have an absolutely continuous bivariate distribution that satisfies BLMP and has exponential marginal distributions. Since absolute continuity is a desirable criterion, Block and Basu proposed the ACBVE where the marginals are mixtures or weighted averages of exponential distributions.