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

2026-03-27

AI Summary

This study constructs a continuous family of inequality measures that unifies the Hoover index and the Gini coefficient. By normalizing a convex combination of mean deviation and the expected absolute pairwise difference, the authors propose a new class of indices satisfying scale invariance and the Pigou–Dalton transfer principle. Under the assumption of a gamma distribution, closed-form expressions are derived using the incomplete gamma function, and an explicit formula for the bias of the plug-in estimator is provided. Monte Carlo simulations demonstrate that both bias and mean squared error of the estimator decline substantially with increasing sample size. Empirical analysis using per capita GDP data confirms the practical utility and flexibility of the proposed framework.

Technology Category

• Game Theory and Economic Paradigms: Fair Division
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty
• Philosophy and Ethics of AI: Bias, Fairness & Equity

Application Category

• Security and Privacy: Large-scale security measurements
• Social Networks and Social Media: Fairness and bias in social network and social media analysis
• Economics, Online Markets and Human Computation: Fairness, privacy, and diversity in economic environments

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.

Problem

• inequality indices
• Hoover index
• Gini coefficient
• unification
• economic inequality

Innovation

• inequality index
• Hoover index
• Gini coefficient
• gamma distribution
• plug-in estimator