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

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• 作者：Kowalczyk ; Teresa
• 关键词：62H20 ; 60E05 ; Copula ; Concentration curves ; Gini separation index ; Grade correlation ; Kendall's tau ; Monotone dependence ; Spearman's rho
• 刊名：Statistics and Probability Letters
• 出版年：2000
• 期刊代码：56_01677152
• 类别：mt
• 出版时间：February 15, 2000
• 卷：46
• 期：4
• 页码：371-379
• 文件大小：108 K

摘要

Two grade measures of monotone dependence, Spearman's ρ^* and Kendall's τ, 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 ρ_max^* and τ_max, defined, respectively, as the maximal values of ρ^* and τ 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 ρ^*(X,Y)=ρ_max^*(X,Y) and τ(X,Y)=τ_max(X,Y). Moreover, another index τ_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 τ_abs and τ_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.