The density of the sample correlations under elliptical symmetry with or without the truncated variance-ratio

The probability density function (pdf) of the sample correlation coefficient, when the associated sample variance-ratio is variously truncated, is given for the bivariate elliptical distribution (elliptical symmetry) with some moments. It is derived that the joint pdf of the sample variance-ratios and correlation matrix with possible truncation under multivariate elliptical symmetry is unchanged irrespective of distinct elliptical distributions. A general condition for transformed sample variances and covariances to have an unchanged pdf under elliptical symmetry is given. Examples satisfying this condition are shown for the intraclass correlation, coefficient alpha and principal component analysis.(c) 2022 Published by Elsevier Inc.