Rank-dependent set-based association measures

Abstract

Association measures capturing rank-dependent stochastic dependence of two random variables are considered. These measures can be seen as special functions on the class of all 2-dimensional copulas. The major part of this work is devoted to the introduction and study of linear rank-dependent association measures $L_G$ based on finite sets $G=\left\{\right(x_1,y_1),\dots ,(x_k,y_k\left)\right\}$ of points from the open unit square $]0,1[{}^2$. Among several examples, we also present discrete approximations of Spearman's ρ and Spearman's footrule ϕ. To capture possibly different importances of points $(x_i,y_i)\in G$, we propose to equip the sets G with a probability P, and consider the P-sets $G_P=\left\{\right(x_1,y_1,p_1),\dots ,(x_k,y_k,p_k\left)\right\}$. Then we study the related association measures $L_{G_P}$. Finally, rank-dependent association measures based on probabilities or special σ-additive measures, not necessarily bounded, are considered.