Numerical Computation of Multivariate Normal Probabilities

Abstract

The numerical computation of a multivariate normal probability is often a difficult problem. This article describes a transformation that simplifies the problem and places it into a form that allows efficient calculation using standard numerical multiple integration algorithms. Test results are presented that compare implementations of two algorithms that use the transformation, with currently available software. KEY WORDS: multivariate normal distribution, Monte-Carlo, adaptive integration. 1 Introduction A problem that arises in many statistics applications is that of computing the multivariate normal distribution function F (a; b) = 1 p j\\Sigmaj(2) m Z b 1 a 1 Z b 2 a 2 ::: Z bm am e \\Gamma 1 2 ` t \\Sigma \\Gamma1 ` d`; where ` = (`1 ; `2 ; :::; `m) t and \\Sigma is an m \\Theta m symmetric positive definite covariance matrix. If, for some i, a i is \\Gamma1 and b i is 1, an appropriate transformation allows the ith variable to be integrated explicitly and reduc...