Saddlepoint Series for Densities

Abstract

In many statistical applications, approximations to the probability that a random variable exceeds a certain threshold value are important. Such approximations are useful, for example, in constructing tests and confidence intervals, and for and calculating p-values. Edgeworth series converge uniformly quickly over the entire possible range of the random variable, when error is measured in an absolute sense. Often times, relative error behavior is more important than absolute error behavior; an error of .005 is of little importance when considering tests of approximate size .05 but is of great importance when considering tests of approximate size .001. Saddlepoint methodology is a method for achieving in many cases uniform bounds on relative error over the range of the distribution. This work was pioneered by Daniels (1954).