Four simple axioms of dependence measures

Tamás F. Móri () and Gábor J. Székely ()
Additional contact information
Tamás F. Móri: ELTE Eötvös Loránd University
Gábor J. Székely: National Science Foundation

Metrika: International Journal for Theoretical and Applied Statistics, 2019, vol. 82, issue 1, No 1, 16 pages

Abstract: Abstract Recently new methods for measuring and testing dependence have appeared in the literature. One way to evaluate and compare these measures with each other and with classical ones is to consider what are reasonable and natural axioms that should hold for any measure of dependence. We propose four natural axioms for dependence measures and establish which axioms hold or fail to hold for several widely applied methods. All of the proposed axioms are satisfied by distance correlation. We prove that if a dependence measure is defined for all bounded nonconstant real valued random variables and is invariant with respect to all one-to-one measurable transformations of the real line, then the dependence measure cannot be weakly continuous. This implies that the classical maximal correlation cannot be continuous and thus its application is problematic. The recently introduced maximal information coefficient has the same disadvantage. The lack of weak continuity means that as the sample size increases the empirical values of a dependence measure do not necessarily converge to the population value.

Keywords: Correlation; Distance correlation; Maximal correlation; Maximal information coefficient; Invariance (search for similar items in EconPapers)
Date: 2019
References: View references in EconPapers View complete reference list from CitEc
Citations:

Downloads: (external link)
http://link.springer.com/10.1007/s00184-018-0670-3 Abstract (text/html)
Access to the full text of the articles in this series is restricted.

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:spr:metrik:v:82:y:2019:i:1:d:10.1007_s00184-018-0670-3

Ordering information: This journal article can be ordered from
http://www.springer.com/statistics/journal/184/PS2

DOI: 10.1007/s00184-018-0670-3

Access Statistics for this article

Metrika: International Journal for Theoretical and Applied Statistics is currently edited by U. Kamps and Norbert Henze

More articles in Metrika: International Journal for Theoretical and Applied Statistics from Springer
Bibliographic data for series maintained by Sonal Shukla () and Springer Nature Abstracting and Indexing ().