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

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. Gini's gamma then 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, i.e., the collection of all square matrices of pairwise generalized Gini's gammas, are derived.