A Conditional Saddlepoint Approximation for Testing Problems

Abstract

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