Link between grade measures of dependence and of separability in pairs of conditional distributions

Two grade measures of monotone dependence, Spearman's [rho]* and Kendall's [tau], can be expressed as weighted averages of monotone Gini separation indices for pairs of conditional distributions of Y on X. This fact is used to show an important property of the measures of absolute dependence [rho]max* and [tau]max, defined, respectively, as the maximal values of [rho]* and [tau] over the set of pairs of all the possible one-to-one Borel-measurable transformations of X and of Y. Namely, if (X,Y) are totally positive of order two (TP2) then [rho]*(X,Y)=[rho]max*(X,Y) and [tau](X,Y)=[tau]max(X,Y). Moreover, another index [tau]abs(X,Y) of absolute dependence is introduced as weighted average of Gini (absolute) separation indices for the pairs of conditional distributions of Y on X. Indices [tau]abs and [tau]max are used to measure the irregularity of dependence. All facts proved in this paper hold for the general case of the mixed discrete-continuous variables.