Matrix compatibility and correlation mixture representation of generalized Gini's gamma

TL;DR

This paper explores the representation of measures of concordance, particularly generalized Gini's gamma, as mixtures of Pearson correlation coefficients of transformed variables, providing bounds for their compatibility matrices.

Contribution

It introduces a novel mixture representation of generalized Gini's gamma using Pearson correlations and derives bounds for compatible matrices of these measures.

Findings

Generalized Gini's gamma can be expressed as a mixture of Pearson correlations.

Bounds for the set of compatible matrices of generalized Gini's gamma are established.

Transformations of variables characterize measures of concordance in terms of Pearson's correlation.

Abstract

Representations of measures of concordance in terms of Pearson' s correlation coefficient are studied. All transforms of random variables are characterized such that the correlation coefficient of the transformed random variables is a measure of concordance. Next, Gini' s gamma is generalized and it is shown that the resulting generalized Gini' s gamma can be represented as a mixture of measures of concordance that are Pearson' s correlation coefficients of transformed random variables. As an application of this correlation mixture representation of generalized Gini' s gamma, lower and upper bounds of the compatible set of generalized Gini' s gamma, which is the collection of all possible square matrices whose entries are pairwise bivariate generalized Gini' s gammas, are derived.