Evaluation of orthant probabilities for singular BI-and trivariate normal distributions

Abstract

This paper presents a method of computing orthant probabilities for singular bivariate and trivariate Normal distributions. Such distributions are intrinsically more complicated to handle than the non- singular multivariate Normal distribution. Indeed, most multivariate analysis texts avoid a detailed discussion of the singular distribution. For the rank-one covariance matrix case, simple and direct methods may be applied. Indeed, the singular bivariate Normal quadrant probabilities are computed by one commercially-available subroutine. However, the rank-two covariance matrix case in trivariate Normal distributions has not apparently been tackled before. Using the property of principal components that they effectively reduce the dimension of the sample space to the rank of the covariance matrix and are themselves independent, the problem of computing octant probabilities for the rank-two co- variance matrix trivariate Normal distribution is essentially reduced to the computation of independent bivariate N...