This paper studies a class of rank-based inequality measures built from linear combinations of expected order statistics. The proposed framework unifies several well-known indices, including the classical Gini coefficient, the $m$th Gini index, extended $m$th Gini index and $S$-Gini index, and also connects to spectral inequality measures through an integral representation. We investigate the finite-sample behavior of a natural U-statistic-type estimator that averages weighted order-statistic contrasts over all subsamples of fixed size and normalizes by the sample mean. A general bias decomposition is derived in terms of components that isolate the effect of random normalization on each rank level, yielding analytical expressions that can be evaluated under broad non-negative distributions via Laplace-transform methods. Under mild moment conditions, the estimator is shown to be asymptotically unbiased. Moreover, we prove exact unbiasedness under gamma populations for any sample size, extending earlier unbiasedness results for Gini-type estimators. A Monte Carlo study is performed to numerically check that the theoretical unbiasednes under gamma populations.

翻译：本文研究了一类基于秩的不等式测度，其由期望次序统计量的线性组合构建而成。所提出的框架统一了若干经典指数，包括传统基尼系数、第$m$阶基尼指数、扩展第$m$阶基尼指数与$S$-基尼指数，并通过积分表示与谱不等式测度建立了联系。我们考察了一种自然U统计型估计量的有限样本性质，该估计量通过对所有固定容量子样本的加权次序统计量对比进行平均，并以样本均值进行归一化。通过分离各秩水平上随机归一化影响的分解项，我们推导出通用的偏差分解式，得到可在广泛非负分布下通过拉普拉斯变换方法计算的解析表达式。在温和矩条件下，该估计量被证明具有渐近无偏性。此外，我们证明了在伽马总体下对任意样本量均具有精确无偏性，这扩展了早期基尼型估计量的无偏性结论。通过蒙特卡洛实验对伽马总体下的理论无偏性进行了数值验证。