Best-possible bounds on the set of copulas with a given value of Gini's gamma

TL;DR

This paper derives the tightest possible bounds on copulas with a fixed Gini's gamma, revealing that these bounds are generally quasi-copulas rather than copulas, unlike bounds for other dependence measures.

Contribution

It establishes the pointwise best-possible bounds for copulas with a given Gini's gamma, highlighting their quasi-copula nature.

Findings

Bounds are not necessarily copulas but quasi-copulas.

Compared to other measures, Gini's gamma bounds are distinct.

Provides a theoretical framework for dependence measure bounds.

Abstract

In this note, pointwise best-possible (lower and upper) bounds on the set of copulas with a given value of the Gini's gamma coefficient are established. It is shown that, unlike the best-possible bounds on the set of copulas with a given value of other known measures such as Kendall's tau, Spearman's rho or Blomqvist's beta, the bounds found are not necessarily copulas, but proper quasi-copulas.