Unbiased estimation in new Gini index extensions under gamma distributions

TL;DR

This paper introduces two new unbiased Gini index estimators for gamma-distributed data, providing exact expectations, assessing finite-sample performance through simulations, and applying them to real income data.

Contribution

It proposes novel unbiased Gini estimators for gamma distributions, extending previous work and deriving their exact expectations.

Findings

Exact expectations of the estimators are derived.

Simulations show good finite-sample performance.

Application to real income data demonstrates practical usefulness.

Abstract

In this paper, we propose two new flexible Gini indices (extended lower and upper) defined via differences between the $i$-th observation, the smallest order statistic, and the largest order statistic, for any $1 \leqslant i \leqslant m$. For gamma-distributed data, we obtain exact expectations of the estimators and establish their unbiasedness, generalizing prior works by [Deltas, G. 2003. The small-sample bias of the gini coefficient: Results and implications for empirical research. Review of Economics and Statistics 85:226-234] and [Baydil, B., de la Pe\~na, V. H., Zou, H., and Yao, H. 2025. Unbiased estimation of the gini coefficient. Statistics & Probability Letters 222:110376]. Finite-sample performance is assessed via simulation, and real income data set is analyzed to illustrate the proposed measures.