APPROXIMATIONS TO MULTIVARIATE NORMAL RECTANGLE PROBABILITIES BASED ON CONDITIONAL EXPECTATIONS

Two new approximations for multivariate normal probabilities for recta ngular regions, based on conditional expectations and regression with binary variables, are proposed. One is a second-order approximation th at is much more accurate but also more numerically time-consuming than the first-order approximation. A third approximation, based on the mo ment-generating function of a truncated multivariate normal distributi on, is proposed for orthant probabilities only. Its accuracy is betwee n the first- and second-order approximations when the dimension is les s than seven and the correlations are not large. All of the approximat ions get worse as correlations get larger. These new approximations of fer substantial improvements on previous approximations. They also com pare favorably with the methods of Genz for numerical evaluation of th e multivariate normal integral. The approximation methods should be es pecially useful within a quasi-Newton routine for parameter estimation in discrete models that involve the multivariate normal distribution.