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

Manuel Úbeda-Flores

TL;DR

This work addresses bounding the set of copulas with a fixed Gini's gamma, $\gamma(C)=t$, by constructing pointwise best-possible lower and upper bounds $\underline{G}_t$ and $\overline{G}_t$. It establishes a symmetry relation $\underline{G}_t(u,v)=v-\overline{G}_{-t}(1-u,v)$ and derives an explicit, region-based form for the upper bound $\overline{G}_t$, with five threshold components $\theta_i$ governing its structure. The corresponding lower bound is $\underline{G}_t(u,v)=v-\overline{G}_{-t}(1-u,v)$. A key finding is that, unlike bounds for other measures such as Kendall's $\tau$, Spearman's $\rho$, or Blomqvist's $\beta$, these gamma-based bounds can be proper quasi-copulas for $-1<t<0$ or $0<t<1$, offering new insights for dependence modeling under a fixed gamma constraint.

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.

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

TL;DR

This work addresses bounding the set of copulas with a fixed Gini's gamma, , by constructing pointwise best-possible lower and upper bounds and . It establishes a symmetry relation and derives an explicit, region-based form for the upper bound , with five threshold components governing its structure. The corresponding lower bound is . A key finding is that, unlike bounds for other measures such as Kendall's , Spearman's , or Blomqvist's , these gamma-based bounds can be proper quasi-copulas for or , offering new insights for dependence modeling under a fixed gamma constraint.

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.
Paper Structure (4 sections, 5 theorems, 46 equations, 5 figures)

This paper contains 4 sections, 5 theorems, 46 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Best-possible bounds when a given value of Gini's gamma is known
  4. Conclusions

Key Result

Proposition 1

Let $C$ be a copula, and suppose $C(a,b)=\theta$, where $(a,b)\in [0,1]^2$ and $W(a,b)\le \theta\le M(a,b)$. Then $\underline{C}_{(a,b),\theta}(u,v)\le C(u,v)\le \overline{C}_{(a,b),\theta}(u,v)$ for all $(u,v)\in[0,1]^2$, where and Since $\underline{C}_{(a,b),\theta}(a,b)=\overline{C}_{(a,b),\theta}(a,b)=\theta$, the bounds, which are copulas, are best-possible.

Figures (5)

  • Figure 1: The respective cases I, II and III (from left to right) for computing the integral \ref{['Intunder']}.
  • Figure 2: The values of the copula $\underline{C}_{(a,b),\theta}$ in case I for computing \ref{['Intunder']}.
  • Figure 3: The respective cases $1,2,3,4$ and $5$ (from left to right) for computing the integral in \ref{['Intunder2']}.
  • Figure 4: The values of the copula $\underline{C}_{(a,b),\theta}$ in case 2 for computing \ref{['Intunder2']}. Note that the inequalities $a-\theta\le 1-a+\theta$, $a-\theta\ge b$ and $1-a+\theta\le a$ must be satisfied in this case.
  • Figure 5: The graph (left) and the level curves (right) of the copula $\overline{G}_0$.

Theorems & Definitions (10)

  • Proposition 1
  • Proposition 2
  • proof
  • Theorem 3
  • proof
  • Remark 1
  • Proposition 4
  • proof
  • Remark 2
  • Proposition 5