On conditional correlations defined by a class of nonlinearly truncated distributions and their applications

Abstract

This article derives and studies several types of conditional correlations. The correlations are obtained by a class of two-piece scale mixture skew-normal distributions. The class is obtained by applying a set of nonlinear constraints to the bivariate scale mixture of normal distributions. The correlations of the class are invariant with respect to the choice of the scale mixing function, however, they are dependent upon the type of the nonlinear truncation. Moreover, their respective upper and lower limits are no longer 1.00 and−1.00. They are useful for the truncated data analysis, the multivariate interdependence methods (such as the principal component analysis and the factor analysis), and the random truncation modelling. Some distributional properties and the Bayesian computation of the correlations are considered when developing necessary theories and providing illustrative examples, respectively. Two applications are also given to demonstrate the usefulness of the conditional correlations in a multivariate analysis.