Distribution of the correlation coefficient for the class of bivariate elliptical models

We consider n pairs of random variables (X 11 ,X 21 ) ,(X 12 ,X 22 ),… (X 1n ,X 2n ) having a bivariate elliptically contoured density of the form\documentclass{article}\pagestyle{empty}\begin{document}$$ K(n)|\Lambda |^{ - n/2} g\left\{ {\sum\limits_1^n {({\bf x}_{1j} - \theta _1 ,\,{\bf x}_{2j} - \theta _2 )\Lambda ^{ - 1} ({\bf x}_{1j} - \theta _1 ,\,{\bf x}_{2j} - \theta _2 )^{\rm T} } } \right\}, $$\end{document}where θ 1 θ 2 are location parameters and Δ = ((λ ik )) is a 2 × 2 symmetric positive definite matrix of scale parameters. The exact distribution of the Pearson product‐moment correlation coefficient between X 1 and X 2 is obtained. The usual case when a sample of size n is drawn from a bivariate normal population is a special case of the abovementioned model.