Open Access
July, 1981 On Nonparametric Measures of Dependence for Random Variables
B. Schweizer, E. F. Wolff
Ann. Statist. 9(4): 879-885 (July, 1981). DOI: 10.1214/aos/1176345528

Abstract

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 $\rho$.

Citation

Download Citation

B. Schweizer. E. F. Wolff. "On Nonparametric Measures of Dependence for Random Variables." Ann. Statist. 9 (4) 879 - 885, July, 1981. https://doi.org/10.1214/aos/1176345528

Information

Published: July, 1981
First available in Project Euclid: 12 April 2007

zbMATH: 0468.62012
MathSciNet: MR619291
Digital Object Identifier: 10.1214/aos/1176345528

Subjects:
Primary: 62E10
Secondary: 62H05

Keywords: copulas , Nonparametric measures of dependence , Renyi's axioms for measures of dependence

Rights: Copyright © 1981 Institute of Mathematical Statistics

Vol.9 • No. 4 • July, 1981
Back to Top