Computation of the quadrivariate and pentavariate normal cumulative distribution functions

This article provides explicit integration rules for the quadrivariate and the pentavariate normal distribution. By analytically reducing the dimension of the problem and simplifying the functions to be integrated, these rules form the basis for a numerical evaluation scheme yielding an observed maximum error in the order of 10− 7 and a computational time of less than 10− 6 s. The implementation is very straightforward as it is based on a classical Gauss–Legendre quadrature. Order statistics are also dealt with.