Partial saddlepoint approximations for transformed means

The full saddlepoint approximation for real valued smooth functions of means requires the existence of the joint cumulant generating function for the entire vector of random variables which are being transformed. We propose a mixed saddlepoint-Edgeworth approximation requiring the existence of a cumulant generating function for only part of the random vector considered, while retaining partially the relative nature of the errors. Tail probability approximations are obtained under conditions which enable the approximations to be used in resampling situations and hence to obtain a result on the relative error of coverage in the case of the bootstrap approximation to the confidence interval for the Studentized mean