JOINT DISTRIBUTIONS OF SOME INDICES BASED ON CORRELATION COEFFICIENTS

Abstract

This chapter presents joint distributions of some indices based on correlation coefficients. One of the most important and commonly used scale-free measures of association is the product-moment correlation coefficient. Most data analytic problems require at least two generalizations of the bivariate product-moment correlation, namely, the partial and the multiple correlations. The chapter presents a general theorem on the asymptotic joint distribution of the determinants of arbitrary correlation matrices of variables. This theorem is the fundamental tool used to obtain the joint distributions of partial, multiple, and partial-multiple correlations. Determinants or elements of the inverse of correlation matrices are involved in the definitions of many indices based on correlations, for example, multiple and partial correlations. Because correlation matrices of normal variates are functions of sample moments, they can have asymptotic normal distributions. The distribution of the determinants of correlation matrices is a starting point for obtaining the distributions of functions of such determinants. In some cases, the asymptotic covariance matrix has a simple, compact form.