# A measure of mutual complete dependence

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## Date

2007-05-25T12:33:09Z

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## 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

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## Keywords

Copulas, Mutual complete dependence, Nonparametric measures of dependence, Sobolev norm