Conditional Saddle-point Approximations for Bivariate Compound Distributions

The majority of existing research that is related to our study aims to explain phenomena in various fields of application that rely on bivariate random variables [1,2]. Although these distributions have attracted some attention in the literature, little research exists on the bivariate compound distribution due to computational difficulties in implementing it. This study introduces the conditional saddle-point approximation method, which is more powerful than other approximation methods, to the bivariate compound distribution in continuous and discrete settings. We discuss conditional approximations for cumulative distribution functions of bivariate compound distributions. Furthermore, examples of continuous and discrete distributions from the bivariate compound truncated Poisson compound class are presented and comparisons between saddle-point approximations and exact calculations show the great accuracy of the saddle-point methods.