Saddlepoint Approximations for Marginal and Conditional Probabilities of Transformed Variables

Abstract

The Lugannani and Rice formula for tail areas in the univariate case has recently been extended to tail areas for marginal distributions and for conditional distributions in certain multivariate settings. However, the results on relative order of errors are given only formally or under the strong continuity assumptions necessary to obtain density approximations. This paper attempts to give a unified treatment of these results for smooth transformations of multivariate means under weaker conditions appropriate to indirect Edgeworth approximations for probabilities.