Approximation Methods for the Bivariate Compound Zero-Truncated Poisson-Gamma Distribution

In certain situations, probability computations can become complex, particularly when dealing with compound distributions. This computational complexity can be simplified by using approximation techniques such as the " saddle -point " approximation. In this paper, the authors have proposed the bivariate compound zero -truncated Poisson -Gamma distribution. This distribution is obtained by compounding the zero -truncated Poisson distribution with independent Gamma variates. To demonstrate the effectiveness of the proposed approach, the authors have provided an illustrative example to showcase the approximate computation of the bivariate compound zerotruncated Poisson- Gamma distribution. Furthermore, an extensive simulation study has been conducted to evaluate the performance of the proposed saddle -point approximation. The results indicate that the proposed saddle -point approximation is an excellent approximation of the distribution function of the bivariate compound zero -truncated Poisson -Gamma distribution which validates the effectiveness of the proposed approach. The high accuracy of the saddle -point approximation method is demonstrated by comparisons among saddle -point approximations, normal approximations, and exact calculations.