Computing Multivariate Normal Probabilities: A New Look

This article describes and compares several numerical methods for finding multivariate probabilities over a rectangle. A large computational study shows how the computation times depend on the problem dimensions, the correlation structure, the magnitude of the sought probability, and the required accuracy. No method is uniformly best for all problems and—unlike previous work—this article gives some guidelines to help establish the most promising method a priori. Numerical tests were conducted on approximately 3,000 problems generated randomly in up to 20 dimensions. Our findings indicate that direct integration methods give acceptable results for up to 12-dimensional problems, provided that the probability mass of the rectangle is not too large (less than about 0.9). For problems with small probabilities (less than 0.3) a crude Monte Carlo method gives reasonable results quickly, while bounding procedures perform best on problems with large probabilities (> 0.9). For larger problems numerical integration ...