Unbiased estimation in new Gini index extensions under gamma distributions, with application to real income data

In this paper, we introduce two flexible extensions of the classical Gini index, referred to as the extended lower and upper Gini indices. The proposed measures are based on the differences between an observation and the minimum and maximum order statistics in samples of size $m\geqslant 2$ and reduce to the classical Gini coefficient when $m=2$. Unlike conventional Gini-type measures, they provide a position-oriented assessment of inequality relative to the lower and upper tails of the distribution. We establish the consistency and asymptotic normality of the proposed estimators under mild regularity conditions. For gamma-distributed populations, we derive exact expressions for their expectations and prove their unbiasedness, thereby extending previous results of [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]. The finite-sample performance of the estimators is investigated through Monte Carlo simulations, and an application to 2023 GDP per capita data from South American countries illustrates the practical usefulness of the proposed measures. The results show that the extended lower and upper Gini indices provide a richer and more informative characterization of inequality than traditional Gini-type measures.