On secant approximations to cumulative distribution functions

Abstract

SUMMARY We investigate the properties of an approximation, called the secant approximation, to the cumulative distribution function where the density is of a broad parametric class. Formulae for higher order terms are derived that give the approximation explicitly in terms of functions that define the density. In all cases examined, the first order secant approximation is monotonic and has the correct limits at both ends of the support, resulting in the important property that the approximation itself is a cumulative distribution function. We give conditions under which the approximation has bounded relative errors throughout the support. The superiority of the secant approximation is illustrated in the particular case of the skew ι family of distributions, also known as Pearson's Type IV family. A collection of formulae useful for the systematic evaluation of these approximations is given in Appendix 2.