Asymptotic Approximations for Multinormal Integrals

Abstract

The problem of deriving approximations for multinormal integrals is examined using results of asymptotic analysis. The boundary of the integration domain given by $g\left(\underset{\text{¯}}{x}\right)=0$ is simplified by replacing $g\left(\underset{\text{¯}}{x}\right)$ by its Taylor expansion at the points on the boundary with minimal distance to the origin. Two approximations which are obtained by using a linear or quadratic Taylor expansion are compared. It is shown that, applying a quadratic Taylor expansion, an asymptotic approximation for multinormal integrals can be obtained, whereas using linear approximations large relative errors may occur.