Authors: Siburg, Karl Friedrich
Stoimenov, Pavel A.
Title: A measure of mutual complete dependence
Language (ISO): en
Abstract: Two random variables X and Y are mutually completely dependent (m.c.d.) if there is a measurable bijection f with P(Y = f(X)) = 1. For continuous X and Y , a natural approach to constructing a measure of dependence is via the distance between the copula of X and Y and the independence copula. We show that this approach depends crucially on the choice of the distance function. For example, the Lp-distances, suggested by Schweizer and Wolff, cannot generate a measure of (mutual complete) dependence, since every copula is the uniform limit of copulas linking m.c.d. variables. Instead, we propose to use a modified Sobolev norm, with respect to which, mutual complete dependence cannot approx- imate any other kind of dependence. This Sobolev norm yields the first nonparametric measure of dependence capturing precisely the two extremes of dependence, i.e., it equals 0 if and only if X and Y are independent, and 1 if and only if X and Y are m.c.d. AMS 2000 subject classifcations: Primary 62E10; secondary 62H20
Subject Headings: Copulas
Mutual complete dependence
Nonparametric measures of dependence
Sobolev norm
Issue Date: 2007-05-25T12:33:09Z
Appears in Collections:Sonderforschungsbereich (SFB) 475

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