Equally Correlated Variates and the Multinormal Integral

Abstract

Ann-dimensional distribution with all correlations equal top,called an equally correlated distribution, can very simply be constructed from a set ofn+ 1 one-dimensional variates; these variates are uncorrelated ifpis non-negative, andnof them are uncorrelated ifpis 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 attest 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.