The Annals of Statistics

On Nonparametric Measures of Dependence for Random Variables

B. Schweizer and E. F. Wolff

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In 1959 A. Renyi proposed a set of axioms for a measure of dependence for pairs of random variables. In the same year A. Sklar introduced the general notion of a copula. This is a function which links an $n$-dimensional distribution function to its one-dimensional margins and is itself a continuous distribution function on the unit $n$-cube, with uniform margins. We show that the copula of a pair of random variables $X, Y$ is invariant under a.s. strictly increasing transformations of $X$ and $Y$, and that any property of the joint distribution function of $X$ and $Y$ which is invariant under such transformations is solely a function of their copula. Exploiting these facts, we use copulas to define several natural nonparametric measures of dependence for pairs of random variables. We show that these measures satisfy reasonable modifications of Renyi's conditions and compare them to various known measures of dependence, e.g., the correlation coefficient and Spearman's $\rho$.

Article information

Ann. Statist. Volume 9, Number 4 (1981), 879-885.

First available in Project Euclid: 12 April 2007

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Zentralblatt MATH identifier


Primary: 62E10: Characterization and structure theory
Secondary: 62H05: Characterization and structure theory

Nonparametric measures of dependence copulas Renyi's axioms for measures of dependence


Schweizer, B.; Wolff, E. F. On Nonparametric Measures of Dependence for Random Variables. Ann. Statist. 9 (1981), no. 4, 879--885. doi:10.1214/aos/1176345528.

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