Bivariate Distributions with Exponential Conditionals

Abstract

Abstract It is frequently easier to visualize conditional distributions of experimental variables rather than joint distributions. In this article we consider the most general class of bivariate distributions such that both sets of conditional densities are exponential. The class proves to be remarkably simple to describe: The joint density must be proportional to exp(- λx - μy - νxy), where the constant of proportionality depends on the classical exponential integral. The joint distribution has marginals that are not exponential and a negative correlation coefficient, except in the special case of independence. After deriving some distributional results, we develop methods for parameter estimation and simulation. A simple method-of-moments estimator appears to give reasonable results. We also briefly discuss generalizations to higher dimensions and to distributions with conditionals in a general exponential family.