On the bias of the Gini estimator: Poisson and geometric cases, a characterization of the gamma family, and unbiasedness under gamma distributions

TL;DR

This paper provides a new theoretical framework for understanding the bias of the Gini coefficient estimator, especially in Poisson, geometric, and gamma distributions, and introduces bias correction methods with improved finite-sample performance.

Contribution

It offers a novel representation of the Gini estimator's expectation using Laplace transforms, characterizes the gamma family via exponential tilting, and develops bias-corrected estimators with demonstrated effectiveness.

Findings

Gini estimator is biased for Poisson and geometric populations.

Unbiasedness of the Gini estimator is confirmed for gamma populations.

Bias-corrected estimators significantly improve finite-sample accuracy.

Abstract

In this paper, we derive a general representation for the expectation of the Gini coefficient estimator in terms of the Laplace transform of the underlying distribution, together with the mean and the Gini coefficient of its exponentially tilted version. This representation leads to a new characterization of the gamma family within the class of nonnegative scale families, based on a stability property under exponential tilting. As direct applications, we show that the Gini estimator is biased for both Poisson and geometric populations and provide an alternative, unified proof of its unbiasedness for gamma populations. By using the derived bias expressions, we propose plug-in bias-corrected estimators and assess their finite-sample performance through a Monte Carlo study, which demonstrates substantial improvements over the original estimator. Compared with existing approaches, our framework highlights the fundamental role of scale invariance and exponential tilting, rather than distribution-specific algebraic calculations, and complements recent results in Baydil et al. (2025) [Unbiased estimation of the gini coefficient. SPL, 222:110376] and Vila and Saulo (2025a,b) [Bias in Gini coefficient estimation for gamma mixture populations. STPA, 66:1-18; and The mth gini index estimator: Unbiasedness for gamma populations. J. Econ. Inequal].