Abstract
Abstract Úbeda-Flores showed that the range of multivariate Spearman’s footrule for copulas of dimension d ≥ 2 is contained in the interval [−1/d, 1], that the upper bound is attained exclusively by the upper Fréchet-Hoeffding bound, and that the lower bound is sharp in the case where d = 2. The present paper provides characterizations of the copulas attaining the lower bound of multivariate Spearman’s footrule in terms of the copula measure but also via the copula’s diagonal section.
Highlights
Spearman’s footrule is a “measure of disarray” (Diaconis and Graham [1]) which assigns to every copula C : [, ]d → [, ] a value φ(C) in the real numbers
Úbeda-Flores showed that the range of multivariate Spearman’s footrule for copulas of dimension d ≥ is contained in the interval [− /d, ], that the upper bound is attained exclusively by the upper FréchetHoe ding bound, and that the lower bound is sharp in the case where d =
The present paper provides characterizations of the copulas attaining the lower bound of multivariate Spearman’s footrule in terms of the copula measure and via the copula’s diagonal section
Summary
Spearman’s footrule is a “measure of disarray” (Diaconis and Graham [1]) which assigns to every copula C : [ , ]d → [ , ] a value φ(C) in the real numbers. The present paper provides characterizations of the copulas attaining the lower bound of multivariate Spearman’s footrule in terms of the copula measure and via the copula’s diagonal section. We show that, for every dimension d ≥ , the lower bound of Spearman’s footrule is sharp and attained by uncountably many copulas.
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