The numerical evaluation of a class of integrals. II

Consider the integral where x 1 , x 2 , …, x N are jointly distributed in a multivariate normal distribution f ( x 1 , x 2 , …, x N ) with ( p ij ) as the correlation matrix. The integral has been expressed in an infinite series of tetrachoric functions for N ≥2. The infinite series is not only complicated, but also is very slowly convergent and is consequently not of much practical use. Plackett (8) obtains a reduction formula for expressing normal integrals in four variates as a finite sum of single integrals of tabulated functions. These integrals have then to be evaluated by a rather awkward numerical quadrature.