Teresa Kowalczyk

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

849

Abstract

The randomized grade regression function of Y on X and two important grade measures of monotone dependence: Spearman's rho and Kendall's tau are expressed as functions of the family of monotone Gini separation indices for pairs consisting of a conditional distribution of Y on X and a marginal distribution of Y. They are also expressed as functions of the family of monotone Gini separation indices for pairs of conditional distributions of Y on X. This is used to show that, for any bivariate distribution which is totally monotone of order two (TM_2), the maximal values of the considered grade measures of dependence over the set of pairs of all possible one-to-one transformations of X and Y are equal to their absolute values for (X,Y). Consequently, the TM_2 distributions behave with respect to the Spearman's rho and Kendall's tau similarly as do the normal distributions with respect to the Pearson correlation coefficient. All facts proved in this paper hold for the general case of mixed discrete-continuous variables.

Key Words and Phrases: copula; concentration curves; Gini separation index; grade correlation; Kendall's tau; monotone dependence; Spearman's rho.