Bias analysis of a linear order-statistic inequality index estimator: Unbiasedness under gamma populations

2026-02-16

AI Summary

This study investigates the finite-sample bias of inequality index estimators based on order statistics, with a focus on unbiasedness under non-negative distributions such as the gamma family. By developing a unified framework encompassing several classical inequality measures, the work proposes a U-statistic-based estimator that averages weighted order statistics over fixed-size subsamples and normalizes by the sample mean. The paper establishes, for the first time, that this estimator is exactly unbiased for gamma populations at any sample size. It further introduces a general bias decomposition technique that isolates the effect of random normalization across different rank levels. Asymptotic unbiasedness is proven under mild moment conditions, and Monte Carlo simulations corroborate the theoretical findings.

Technology Category

• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty
• Machine Learning: Calibration & Uncertainty Quantification
• Computer Vision: Bias, Fairness & Privacy

Application Category

• Social Networks and Social Media: Fairness and bias in social network and social media analysis
• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking
• Economics, Online Markets and Human Computation: Fairness, privacy, and diversity in economic environments

Abstract

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.

Problem

• bias analysis
• inequality index
• order statistics
• unbiasedness
• gamma distribution

Innovation

• order statistics
• inequality index
• U-statistic
• bias decomposition
• gamma distribution