The Numerical Evaluation of Certain Multivariate Normal Integrals

Abstract

As has been noted by several authors, when a multivariate normal distribution with correlation matrix $\{\rho_{ij}\}$ has a correlation structure of the form $\rho_{ij} = \alpha_i \alpha_j (i \neq j)$, where $-1 \leqq \alpha_i \leqq + 1$, its c.d.f. can be expressed as a single integral having a product of univariate normal c.d.f.'s in the integrand. The advantage of such a single integral representation is that it is easy to evaluate numerically. In this paper it is noted that the $n$-variate normal c.d.f. with correlation matrix $\{\rho_{ij}\}$ can always be written as a single integral in two ways, with an $n$-variate normal c.d.f. in the integrand and the integration extending over a doubly-infinite range, and with an $(n - 1)$-variate normal c.d.f. in the integrand and the integration extending over a singly-infinite range. We shall show that, for certain correlation structures, the multivariate normal c.d.f. in the integrand factorizes into a product of lower-order normal c.d.f.'s. The results may be useful in instances where these lower-order integrals are tabulated or can be evaluated. One important special case is $\rho_{ij} = \alpha_i \alpha_j$, previously mentioned. Another is $\rho_{ij} = \gamma_i/\gamma_j (i < j)$, where $|\gamma_i| < |\gamma_j|$ for $i < j$. Some applications of these two special cases are given.