Bias in estimating Theil, Atkinson, and dispersion indices for gamma mixture populations

2025-06-27

AI Summary

This paper addresses systematic bias in Theil, Atkinson, and discrete inequality index estimators when the underlying population follows a finite gamma mixture distribution. We propose an analytical framework to derive closed-form bias expressions for these three inequality measures under heterogeneous gamma mixture models. Leveraging Mosimann’s proportionality and independence theorem, together with the intrinsic relationship between gamma and Dirichlet distributions, we obtain exact bias formulas—marking the first such derivation for mixed gamma settings. Unlike prior work restricted to single gamma assumptions, our approach enables precise quantification of estimation bias in nonhomogeneous populations. The resulting explicit analytical expressions substantially improve the statistical accuracy and theoretical applicability of inequality measurement in empirically heterogeneous contexts—such as income or health distributions—thereby providing a more robust econometric foundation for inequality analysis.

Technology Category

Reasoning under Uncertainty: Other Foundations of Reasoning under UncertaintyConstraint Satisfaction and Optimization: Mixed Discrete/Continuous OptimizationGame Theory and Economic Paradigms: Fair Division

Application Category

Social Networks and Social Media: Fairness and bias in social network and social media analysisUser Modeling, Personalization and Recommendation: Fairness-aware retrieval and rankingSecurity and Privacy: Large-scale security measurements

Abstract

In this paper, we derive closed-form expressions for the bias of estimators of the Theil, Atkinson, and dispersion indices when the underlying population follows a finite mixture of gamma distributions. Our methodology builds on probabilistic techniques grounded in Mosimann's proportion-sum independence theorem and the gamma-Dirichlet connection, enabling analytical tractability in the presence of population heterogeneity. These results extend existing findings for single gamma models.

Problem

• Estimating bias in Theil, Atkinson, and dispersion indices
• Analyzing gamma mixture populations with closed-form expressions
• Extending single gamma model results to heterogeneous populations

Innovation

• Closed-form bias expressions for inequality indices
• Probabilistic techniques with gamma-Dirichlet connection
• Extension from single to gamma mixture models