Two expansions for the quadrivariate normal integral

Let Xl, X2, X3 and X4 obey a quadrivariate niormal distribution with zero means, aind let P4 be the value of the quadrivariate normal integral, i.e. the probability that Xl, X2, X3 and X4 are simultaneously positive. The generalized tetrachoric series for P4, as given by Aitken (unpublished), Kendall (1941, 1945) (see also Kendall & Stuart, 1958, pp. 350-4), and Moran (1948), are not well suited for computation. For the case in which all six correlation coefficients are equal, the series has been sumlmed approximately by McFadden (1956). For special numerical values of the correlation matrix, exact results for P4 have been given by Schlafli (1858, 1860), Anis & Lloyd (1953), and Plackett (1954). Methods for numerical integration in more general cases have been given by Plackett (1954), Ihm (1959), and John (1959). Numerical methods for integration when all the correlation coefficients are equal have been provided by Ruben (1954) and Moran (1956). In this paper we present two series expansions for P4 which are well suited for computation. The first case occurs when X1, X2, X3 and X4 are successive measurements from a stationary Gaussian Markov process (with zero mean), or, equivalently, when the inverse of the correlation matrix has zero elements except on the main diagonal and immediately adjacent to it. The second case occurs when the correlation matrix itself has zero elements except on the main diagonal and adjacent to it. We shall then show that these two cases are related by a simple transformation. In a previous paper, McFadden (1955, eq. (39)) has expressed P4 in terms of the various product moments of the set of random variables Yi which assume the sign of the normal variables Xi. Let Y.=1 when X. O' ---1 when Xi< Oj (i= 1,2,3,4). (1)