Equally Correlated Variates and the Multinormal Integral

Summary An n-dimensional distribution with all correlations equal to p, called an equally correlated distribution, can very simply be constructed from a set of n + 1 one-dimensional variates; these variates are uncorrelated if p is non-negative, and n of them are uncorrelated if p is negative. From this, a general relation is deduced between any problem concerning an equally correlated distribution and an equivalent problem for uncorrelated variates. Applications are (i) to a t test due to Walsh (1947) and Halperin (1951); (ii) to Moran’s (1956) result for the probability that all the variates of an equally positively correlated normal distribution are positive and to a generalization of this; (iii) to the approximation of sampling distributions under randomization to the results of standard normal theory.