A General Method for Approximating Tail Probabilities
提出一种基于微分方程的G(m)n变换，能生成易计算且高精度的尾部概率近似函数，适用于多种分布，包括极端尾部。
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 show the high accuracy of this method even with small values of n (n = 1, 2, 3). In many cases satisfactory approximation functions can be obtained that are sufficiently simple to be calculated on a hand calculator.