Two expansions for the quadrivariate normal integral
Abstract
Purdue University 1. INTRODUCTION Let Xv Xt, X3 and Xt obey a quadrivariate normal distribution with zero means, and let P4 be the value of the quadrivariate normal integral, i.e. the probability that Xlt X2, X3 and Xt 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 The generalized tetrachoric series for P4, as given by Aitken (unpublished), Kendall (1941, 1945) (see also Kendall & Stuart, 1958, pp. 350-4), and Moran (1948), are not well suited for computation.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 summed 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 Xlt X2, X3 and X4 are successive measurements from a stationary Gaussian Markov process (with zero mean), or, equivalently, when