Unbiased estimation in new Gini index extensions under gamma distributions

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.