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New multivariate Gini's indices

Marco Capaldo, Jorge Navarro

Abstract

The Gini's mean difference was defined as the expected absolute difference between a random variable and its independent copy. The corresponding normalized version, namely Gini's index, denotes two times the area between the egalitarian line and the Lorenz curve. Both are dispersion indices because they quantify how far a random variable and its independent copy are. Aiming to measure dispersion in the multivariate case, we define and study new Gini's indices. For the bivariate case we provide several results and we point out that they are "dependence-dispersion" indices. Covariance representations are exhibited, with an interpretation also in terms of conditional distributions. Further results, bounds and illustrative examples are discussed too. Multivariate extensions are defined, aiming to apply both indices in more general settings. Then, we define efficiency Gini's indices for any semi-coherent system and we discuss about their interpretation. Empirical versions are considered in order as well to apply multivariate Gini's indices to data.

New multivariate Gini's indices

Abstract

The Gini's mean difference was defined as the expected absolute difference between a random variable and its independent copy. The corresponding normalized version, namely Gini's index, denotes two times the area between the egalitarian line and the Lorenz curve. Both are dispersion indices because they quantify how far a random variable and its independent copy are. Aiming to measure dispersion in the multivariate case, we define and study new Gini's indices. For the bivariate case we provide several results and we point out that they are "dependence-dispersion" indices. Covariance representations are exhibited, with an interpretation also in terms of conditional distributions. Further results, bounds and illustrative examples are discussed too. Multivariate extensions are defined, aiming to apply both indices in more general settings. Then, we define efficiency Gini's indices for any semi-coherent system and we discuss about their interpretation. Empirical versions are considered in order as well to apply multivariate Gini's indices to data.
Paper Structure (6 sections, 101 equations, 6 figures, 2 tables)

This paper contains 6 sections, 101 equations, 6 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Background and notations
  3. Bivariate Gini's indices
  4. Bounds and inequalities
  5. Efficiency Gini's indices in systems
  6. Examples and simulations

Figures (6)

  • Figure 1: Plots and areas referring to Eq. (\ref{['eq:bivariateGindex, ID, uniform']}) for the diagonal section $\delta$ in the independent case (black), $M$ (red) and $W$ (blue) and FGM copulas for $\theta=1,0.5$ (orange) and $-1,-0.5$ (green).
  • Figure 2: Plots and areas referring to Eq. (\ref{['eq:bivariateGindex, ID, uniform']}) for the diagonal section $\delta$ of Clayton copulas (left-hand-side) for $\theta=1,2,5,10,20$ (orange) and $\theta=-0.2,-0.4,-0.6,-0.8$ (green), while Frank copulas (right-hand-side) for $\theta=1,2,5,10,20$ (orange) and $\theta=-1,-2,-5,-10,-20$ (green).
  • Figure 3: Plots and areas referring to Eq. (\ref{['eq:bivariateGindex, ID, exponential']}) for $\hat{\delta}(u)/u$ in the exponential independent case (black), $M$ (red) and $W$ (blue) and FGM copulas for $\theta=1,0.5$ (orange) and $-1,-0.5$ (green).
  • Figure 4: Simulated ordered data obtained from the Clayton copula in Example \ref{['ex1']} with standard uniform (left-hand-side) or exponential (right-hand-side) distributions.
  • Figure 5: Simulated ordered data obtained from the Frank copula in Example \ref{['ex2']} with standard uniform (left-hand-side) or exponential (right-hand-side) distributions.
  • ...and 1 more figures