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Unifying the Hoover and Gini indices: Analytical, bias, and computational aspects

Roberto Vila, Helton Saulo, Felipe Quintino

Abstract

We propose a new family of inequality indices that bridges the Hoover index and the Gini coefficient. The measure is defined as the normalized expected absolute value of a convex combination of deviations from the mean and pairwise differences, providing a continuous interpolation between these two classical indices. We establish key theoretical properties, including scale invariance, boundedness, continuity, and compliance with the Pigou-Dalton transfer principle. Analytical representations are derived, allowing explicit evaluation under gamma distributions and leading to closed-form expressions involving incomplete gamma functions. From a statistical perspective, we study the plug-in estimator, obtaining a general expression for its expectation and explicit formulas for its bias under gamma populations. Simulation results indicate good finite-sample performance, with decreasing bias and mean squared error as the sample size increases. An empirical application to GDP per capita data illustrates the practical usefulness of the proposed index as a flexible tool for inequality analysis.

Unifying the Hoover and Gini indices: Analytical, bias, and computational aspects

Abstract

We propose a new family of inequality indices that bridges the Hoover index and the Gini coefficient. The measure is defined as the normalized expected absolute value of a convex combination of deviations from the mean and pairwise differences, providing a continuous interpolation between these two classical indices. We establish key theoretical properties, including scale invariance, boundedness, continuity, and compliance with the Pigou-Dalton transfer principle. Analytical representations are derived, allowing explicit evaluation under gamma distributions and leading to closed-form expressions involving incomplete gamma functions. From a statistical perspective, we study the plug-in estimator, obtaining a general expression for its expectation and explicit formulas for its bias under gamma populations. Simulation results indicate good finite-sample performance, with decreasing bias and mean squared error as the sample size increases. An empirical application to GDP per capita data illustrates the practical usefulness of the proposed index as a flexible tool for inequality analysis.

Paper Structure

This paper contains 10 sections, 6 theorems, 62 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Properties of the index
  3. An auxiliary lemma
  4. Expectation of the estimator $\widehat{I}_\lambda$
  5. Bias of the estimator $\widehat{I}_\lambda$ under gamma populations
  6. Monte Carlo simulation study
  7. Evaluating the Bias and MSE of estimates
  8. Comparison with the competing index $J_\lambda$
  9. Empirical application
  10. Concluding remarks

Key Result

Proposition 2.1

The index $I_\lambda$ of a non-negative, non-degenerate random variable $X$ with finite mean $\mathbb{E}[X]=\mu>0$ can be expressed as where $F_X$ denotes the distribution function of $X$ , $\overline{F}_X(x)=1-F_X(x)$, and $F_X(x^-)=\lim_{u\uparrow x}F_X(u)=\mathbb{P}(X<x)$.

Figures (2)

  • Figure 1: Bias of the estimators $\widehat{I}_\lambda$ (black) and $\widehat{J}_\lambda$ (red) under Gamma$(\alpha,1)$ distributions, based on $R=1000$ Monte Carlo replications, for $\lambda\in\{0.25, 0.5, 0.75\}$.
  • Figure 2: Path of $\widehat{I}_\lambda$ as a function of $\lambda$, with Hoover and Gini as limiting cases.

Theorems & Definitions (25)

  • Proposition 2.1
  • proof
  • Proposition 2.2
  • proof
  • Remark 2.3
  • Remark 2.4
  • Remark 2.5
  • Remark 2.6
  • Remark 2.7
  • Lemma 3.1
  • ...and 15 more