ABSTRACT
This chapter describes the most direct derivation of the saddlepoint approximation, by inversion of the cumulant generating function. The advantages of the saddlepoint approximation over the F. Y. Edgeworth series are mainly connected with accuracy. In the Edgeworth system of approximation, the carrier measure is taken as constant throughout the sample space. Thus the whole burden of approximation lies on the invariant series approximation. Numerical integration is one answer, but this is often cumbersome. A second argument against the saddlepoint approximation in favour of the Edgeworth series is that in order to compute the saddlepoint, it is necessary to have an explicit formula for the cumulant generating function. In short, the Edgeworth series is often easier to use in practice but is usually inferior in terms of accuracy, particularly in the far tails of the distribution.
