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

TL;DR

This paper introduces a new family of inequality indices that unify the Hoover and Gini measures, with theoretical properties, explicit formulas, and practical applications demonstrated through simulations and GDP data.

Contribution

It develops a continuous interpolation between Hoover and Gini indices, providing analytical, bias, and computational insights with explicit formulas and empirical validation.

Findings

Explicit formulas for the new index under gamma distributions.

Bias and expectation formulas for the plug-in estimator.

Simulation results show decreasing bias with larger samples.

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.