Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The W-footrule coefficient: A copula-based measure of countermonotonicity

Enrique de Amo, David García-Fernández, Manuel Úbeda-Flores

Abstract

We introduce the $W$-footrule coefficient $Φ_C$, a copula-based coefficient of negative association defined as the $L^1$-distance to the countermonotonic copula $W$. We prove that Gini's gamma admits the decomposition $γ_C = \frac{2}{3}(\varphi_C+Φ_C)$, linking it to Spearman's footrule $\varphi_C$. A rank-based estimator is introduced, with its strong consistency and asymptotic normality established via the functional delta method. Monte Carlo simulations confirm the estimator's finite-sample validity and its sensitivity to negative dependence structures.

The W-footrule coefficient: A copula-based measure of countermonotonicity

Abstract

We introduce the -footrule coefficient , a copula-based coefficient of negative association defined as the -distance to the countermonotonic copula . We prove that Gini's gamma admits the decomposition , linking it to Spearman's footrule . A rank-based estimator is introduced, with its strong consistency and asymptotic normality established via the functional delta method. Monte Carlo simulations confirm the estimator's finite-sample validity and its sensitivity to negative dependence structures.
Paper Structure (6 sections, 5 theorems, 67 equations, 1 table)

This paper contains 6 sections, 5 theorems, 67 equations, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The $W$-footrule coefficient
  4. The sample version
  5. Simulation study
  6. Conclusions

Key Result

theorem 1

Let $(X_i,Y_i)$, $i=1,\ldots,n$, be an i.i.d. sample from a continuous bivariate distribution with copula $C$. Then $\widehat{\Phi}_n \xrightarrow{\mathrm{a.s.}} \Phi_C$ as $n\to\infty.$

Theorems & Definitions (12)

  • theorem 1
  • proof
  • theorem 2
  • proof
  • remark 1
  • proposition 1
  • proof
  • definition 1
  • proposition 2
  • proof
  • ...and 2 more