Saddlepoint approximation for the studentized mean, with an application to the bootstrap

A saddlepoint technique is used to approximate to the density and tail probability of the studentized mean of a random sample. The motivation was to replace bootstrapping of the studentized mean in the way Davison & Hinkley (1988) used the saddlepoint approximation for the unstudentized mean. The method involves first obtaining a bivariate saddlepoint approximation, then, after a nonlinear transformation, integrating out an unwanted variable either numerically or by a Laplace approximation. The tail probability is similarly evaluated either by a further numerical integration or by a Laplace approximation of the Temme type. Two difficulties arise. (i) The nonlinearity of the transformation may result in Laplace approximations failing in the tail when the sample is not large. But numerical integration always works. (ii) In the bootstrap application the saddlepoint approximation may itself break down when the data set contains an outlier.