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

In this paper, we introduce a novel flexible Gini index, referred to as the extended mth Gini index, which is defined through ordered differences between the jth and kth order statistics within subsamples of size m, for indices satisfying 1 ≤ j < k ≤ 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 [1], [2], and [3]. 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.