A conditional saddlepoint approximation for testing problems

A saddlepoint approximation is provided for the distribution function of on e M statistic conditional on another M statistic. Many interesting statisti cs based on dependent quantities (e.g., spacings, multinomial frequencies, rank differences) can be expressed in terms of independent identically dist ributed random variables conditioned on their sum, so that this conditional saddlepoint approximation yields accurate approximations for the distribut ion of such statistics. This saddlepoint approximation can also be used in conditional testing, where nuisance parameters are eliminated by conditioni ng on sufficient statistics.