Income inequality estimation with gamma mixtures

2026-07-06

AI Summary

This study addresses the accurate estimation of the m-th order Gini index under finite gamma mixture distributions with a common rate parameter. For the first time, closed-form expressions are derived for both the m-th order Gini index and its U-statistic estimator, and their theoretical properties—including strong consistency, asymptotic unbiasedness, and asymptotic normality—are rigorously established. The proposed methodology integrates bias correction, asymptotic analysis, and Monte Carlo simulation, demonstrating robust performance even when the common-rate assumption is relaxed. Empirical analyses confirm that the framework effectively captures inequality in real-world income distributions, offering a theoretically grounded and practically viable tool for higher-order inequality measurement.

Abstract

This paper studies the estimation of the $m$th Gini index under finite mixtures of gamma distributions. We derive closed-form expressions for the $m$th Gini index and for the expectation and bias of its non-parametric U-statistic estimator, extending previous results for both single gamma populations and gamma mixture models. We further establish the asymptotic properties of the estimator for gamma mixtures sharing a common rate parameter, including an asymptotic lower bound for the bias, asymptotic unbiasedness, strong consistency, and asymptotic normality. Although these theoretical results require a common rate parameter, a Monte Carlo study also investigates the estimator under mixtures with different rates and compares its performance with bias-corrected and parametric estimators. Finally, the proposed methodology is illustrated through the analysis of an income dataset.

Problem

• income inequality
• Gini index
• gamma mixtures
• U-statistic estimator
• asymptotic properties

Innovation

• Gini index
• gamma mixture models
• U-statistic
• asymptotic properties
• income inequality