Relative Error of Scaled Poisson Approximation via Stein’s Method

A scaled Poisson approximation has been shown to offer an accurate, yet more conservative approximation to a weighted sum of independent Poisson random variables. The approximation has been widely applied across various domains, yet its theoretical accuracy, especially regarding tail distributions, remains unexplored. In this paper, we address this gap by employing a modified Stein-Chen method to quantify the approximation’s precision. Specifically, we establish a moderate deviation bound on the relative error for the tail distributions using a modified Stein-Chen method. Our results also extend to evaluating the relative error in the tail distributions when approximating compound Poisson distributions.