A General Method for Approximating Tail Probabilities
Abstract A new transformation referred to as the G (m) n -transformation is introduced to establish a general methodology for finding functions that are easy to evaluate and give very good approximations to tail probabilities. The transformation is based only on a general class of differential equations that include the specified density function in the solution set, at least asymptotically. As a result this new method applies to a broad class of distributions. Not only is the method general, it is also accurate. Unlike many other approximation methods, the approximation functions produced maintain their high degree of accuracy even in the extreme tails, making them also suitable for extreme tail probability approximation. In this article the method is applied to the Pearson family members, the inverse Gaussian, and the ratio of χ 2 and lognormal that do not belong to the Pearson family. Furthermore, it is applicable in the case of all of the standard noncentral distributions. Tables are included that sho...