zbMATH — the first resource for mathematics

Extremes of nonexchangeability. (English) Zbl 1110.62071
Summary: For identically distributed random variables \(X\) and \(Y\) with joint distribution function \(H\), we show that the supremum of \(|H(x,y)-H(y,x)|\) is \(1/3\). Using copulas, we define a measure of non-exchangeability, and study maximally non-exchangeable random variables and copulas. In particular, we show that maximally non-exchangeable random variables are negatively correlated in the sense of Spearman’s rho.

62H05 Characterization and structure theory for multivariate probability distributions; copulas
62H20 Measures of association (correlation, canonical correlation, etc.)
Full Text: DOI
[1] Galambos J (1982) Exchangeability, in Encyclopedia of Statistical Sciences, Vol. 2, Kotz, S and Johnson NL, editors. John Wiley & Sons, New York, 573–577 · Zbl 0505.62027
[2] Lancaster HO (1963) Correlation and complete dependence of random variables. Ann. Math. Statist. 34, 1315–1321 · Zbl 0121.35905 · doi:10.1214/aoms/1177703867
[3] Nelsen, RB (1999) An Introduction to Copulas. Springer, New York · Zbl 0909.62052
[4] Schweizer, B, Wolff, EF (1981) On nonparametric measures of dependence for random variables. Ann. Statist. 9, 870–885 · Zbl 0468.62012 · doi:10.1214/aos/1176345528
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.