Representations of multivariate normal distributions with special correlation structures

Abstract

The evaluation of multivariate normal probability integrals has led several authors to reductions of certain integrals for special cases of the correlation matrix (ρij) of jointly normal variates. This reduction is obtained by representing the original normal variates as an appropriate linear combination of a larger set of variates. One important special case is ρij=+αiαj (i≠j), where -1≦αi≦1, which as Gupta (1963) noted, has been periodically and independently rediscovered. It is the purpose of this paper (1) to show that an analogous representation holds true for ρij = =−αiαj (i≠j), provided that (2) to generalize the representations for ρij=+αiαj and ρij=−αiαj; and (3) to apply these representations to integrals for equi-coordinate and orthant probabilities. This application provides a method for finding the distribution of the largest of a set of equi-correlated normal variates. The method requires only the evaluation of a simple integral of Grubbs (1950) function which represents the distribution of the largest deviation from the sample mean. For ρ<0, this result is much simpler for computation than the only currently feasible method presented by Hoffman and Saw (1975).