A General Method for the Approximation of Tail Areas

Abstract

For a density function $f(x)$, the tail area $\alpha(x) = \int^\infty_x f(x) dx,$ may be approximated by $\hat{\alpha}(x) = \frac{f(x)}{g(x)} \cdot (K - 1)^{-1}\cdot\big\{1 + \frac{1}{2} \big(\frac{g'(x)}{g^2(x)} - (K)\big)\big\},$ where $g(x) = f(x)/f'(x)$, and $K = \lim_{x\rightarrow\infty} \{g'(x)/g^2(x)\}$. The formula requires only one constant and three function evaluations; $g$ and $g'$ are typically elementary functions. Such approximations are useful for programmed calculators or very small computers where only a few constants can be stored. The accuracy of the approximation is calculated for some common distributions. The approximation is very accurate for a large class of distributions.