Matrix compatibility and correlation representation of Gini's gamma

Representation of measures of concordance in terms of Pearson's correlation coefficient is studied. We first characterize the transforms such that the correlation coefficient between the transformed random variables is a measure of concordance. Our characterization improves upon that of Hofert and Koike (2019), and covers non-continuous transforms which lead to, for example, Blomqvist's beta. We then generalize Gini's gamma, and show that the generalized Gini's gamma can be represented by a mixture of measures of concordance written as the Pearson's correlation coefficients between transformed random variables. As an application of the correlation mixture representation, we derive lower and upper bounds of the compatible set of generalized Gini's gamma, that is, the collection of all possible square matrices whose entries are pairwise bivariate generalized Gini's gammas.