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

Teresa Kowalczyk

Statistics & Probability Letters, 2000, vol. 46, issue 4, 371-379

Abstract: 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.

Keywords: Copula; Concentration; curves; Gini; separation; index; Grade; correlation; Kendall's; tau; Monotone; dependence; Spearman's; rho (search for similar items in EconPapers)
Date: 2000
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Citations: View citations in EconPapers (3)

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