Conditional Saddle-point Approximations for Bivariate Compound Distributions

The majority of existing research 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. Because of the computational difficulties in implementing such a distribution, research on it is limited. This study aims to use the saddle-point approximation method, which is more powerful than other methods. This article introduces conditional saddle-point approximations to the bivariate compound distribution in continuous and discrete settings. We discuss conditional approximations for cumulative distribution functions of bivariate compound distributions. Examples of continuous and discrete distributions from the bivariate compound truncated Poisson compound class are presented. Comparisons between saddle-point approximations and the exact calculations show the great accuracy of the saddle-point methods.