Approximations to the distribution of sum of independent non-identically gamma random variables

Calculating the sum of independent non-identically distributed random variables is necessary in the scientific field. Computing the probability of the corresponding significance point is important in cases that have a finite sum of random variables. However, it is difficult to evaluate this probability when the number of random variables increases. Under these circumstances, consideration of a more accurate approximation of the distribution function is extremely important. A saddlepoint approximation is performed using upper probabilities from the distribution of the sum of independent non-identically gamma random variables under finite sample sizes. In this study, we compared the results from a saddlepoint approximation to those from normal and moment-based approximations to identify the most appropriate method to use for the distribution function.