An unbiased estimator of a novel extended Gini index for gamma distributed populations

TL;DR

This paper introduces a new unbiased estimator for an extended Gini index tailored for gamma distributed populations, with theoretical derivations, simulation validation, and real data application.

Contribution

It proposes a novel extended Gini index and derives a closed-form unbiased estimator for gamma distributions, extending previous work.

Findings

Estimator is unbiased under gamma distribution

Monte Carlo simulations confirm finite-sample unbiasedness

Application to GDP data demonstrates practical utility

Abstract

In this paper, we introduce a novel flexible Gini index, referred to as the extended Gini index, which is defined through ordered differences between the $j$th and $k$th order statistics within subsamples of size $m$, for indices satisfying $1 \leqslant j \leqslant k \leqslant m$. We derive a closed-form expression for the expectation of the corresponding estimator under the gamma distribution and prove its unbiasedness, thereby extending prior findings by \cite{Deltas2003}, \cite{Baydil2025}, and \cite{Vila2025}. A Monte Carlo simulation illustrates the estimator's finite-sample unbiasedness. A real data set on gross domestic product (GDP) per capita is analyzed to illustrate the proposed measure.