This paper presents a saddlepoint approximation to the cumulative distribution function of a random vector. The proposed approximation has accuracy comparable to that of existing expansions valid in twodimensions, and may be applied to random vectors of arbitrary length, subject only to the requirement that the distribution approximated either have a density or be confined to a lattice, and have a cumulant generating function. The result is derived by directly inverting the multivariate moment generating function. The result is applied to sufficient statistics from a regression model with exponential errors, and compared to an existing method in two dimensions. The result is also applied to multivariate inference from a data set arising from a case-control study of endometrial cancer.
"Multivariate saddlepoint tail probability approximations." Ann. Statist. 31 (1) 274 - 286, Februrary 2003. https://doi.org/10.1214/aos/1046294465