Asymptotic approximations for multivariate integrals with an application to multinormal probabilities

The efficient computation of multinormal integrals is an important problem of multivariate statistics. In this paper it is shown, that using methods of asymptotic analysis, asymptotic expansions for multinormal integrals can be obtained. These results are an extension of a result obtained by Ruben (1964, J. Res. Nat. Bur. Standards B68, No. 1 3-11). While the approximations of Ruben are valid only for domains bounded by hyperplanes, the results given here also apply to domains with nonlinear boundaries.