A General Method for the Approximation of Tail Areas

For a density function f (x), the tail area (x) = ^ₓ f (x) dx, may be approximated by (x) = f (x) g (x) (K - 1) ^-1\1 + 1{2 (g' (x) g² (x) - (K) ) \}, where g (x) = f (x) /f' (x), and K = ₗ \g' (x) /g² (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.