Spearman's footrule and Gini's gamma: Local bounds for bivariate copulas and the exact region with respect to Blomqvist's beta

TL;DR

This paper derives local bounds for Spearman's footrule and Gini's gamma measures of association in bivariate copulas, filling a gap in understanding their behavior and relationships with other measures like Blomqvist's beta.

Contribution

It provides the first derivation of local bounds for Spearman's footrule and Gini's gamma, including their relation to Blomqvist's beta, expanding copula analysis tools.

Findings

Bounds are quasi-copulas, not copulas, for certain measure values.

Relations between Spearman's footrule, Gini's gamma, and Blomqvist's beta are established.

The bounds enhance local dependence analysis in data exploration.

Abstract

Copulas are becoming an essential tool in analyzing data thus encouraging interest in related questions. In the early stage of exploratory data analysis, say, it is helpful to know local copula bounds with a fixed value of a given measure of association. These bounds have been computed for Spearman's rho, Kendall's tau, and Blomqvist's beta. The importance of another two measures of association, Spearman's footrule and Gini's gamma, has been reconfirmed recently. It is the main purpose of this paper to fill in the gap and present the mentioned local bounds for these two measures as well. It turns out that this is a quite non-trivial endeavor as the bounds are quasi-copulas that are not copulas for certain values of the two measures. We also give relations between these two measures of association and Blomqvist's beta.