Orthant Probabilities for the Quadrivariate Normal Distribution

Abstract

Let $x_1, x_2, x_3, x_4$ be jointly distributed with a quadrivariate normal distribution with mean 0, and correlation matrix $\{\rho_{ij}\}$. The orthant probability, i.e. the probability that all the $x_i$'s will be simultaneously positive, is not, in general, given by a closed expression; but is easily computed in a special class of cases, here called orthoscheme probabilities. It is explicitly shown how the general orthant probability can be expressed as a linear combination of six orthoscheme probabilities. Orthoscheme probabilities have been tabulated by the author and instructions for the use of this table [1] are given. In addition, an abridged table is appended.