Exact sampling distribution of the general case sample correlation matrix

Abstract

The distribution of the sample correlation matrix when the sample comes from a real multivariate normal population and when the population covariance matrix is diagonal or when the population variables are independently distributed is available in the literature. In the general case, that is, when the covariance matrix in the normal population is a general real positive definite matrix, the distribution of the sample correlation matrix is not available in a compact computable form in the literature. Some scattered results on some aspects of the general case are available, but the representations are not easily tractable. This article deals with a covariance structure coming from a real matrix-variate gamma when the scale parameter matrix is a general real positive definite matrix. General procedures for deriving asymptotic chi-square and asymptotic normal for some general moment structures are also presented in this article. The density in some particular cases are given explicitly. Graphs are also provided in the particular cases.