Unbiased estimation of normalized scale-invariant indices under the gamma distribution

2026-06-21

AI Summary

This study addresses the lack of a general unbiased estimator for normalized scale-invariant indices—such as the Gini coefficient and entropy-based measures—under gamma-distributed populations. The authors construct such indices using homogeneous functions and leverage the independence between the sum of gamma variables and their Dirichlet-normalized proportions to develop unbiased estimators via U-statistics. This approach establishes, for the first time, a unified framework for unbiased estimation of any normalized scale-invariant index under gamma models. Theoretical analysis and Monte Carlo simulations demonstrate the robustness of the proposed estimators even under generalized gamma distributions. Empirical application to per capita GDP data across the Americas shows that the estimators perform consistently well across various indices and scenarios.

Abstract

We introduce a broad class of normalized scale-invariant indices (NPRIs) generated by homogeneous functions and encompassing several well-known measures, including the Gini coefficient, generalized Gini indices, entropy-based measures, and variability indices. Explicit expressions are obtained for these indices under gamma populations. Exploiting the independence between the total sum and the associated Dirichlet proportions, we derive a simple unbiased estimator based on a U-statistic. The resulting estimator is shown to be unbiased for any NPRI when the underlying population follows a gamma distribution. Several examples are provided to illustrate the general theory. A Monte Carlo simulation study is carried out that shows the good performance of the unbiased estimator in several scenarios of index choices. We also present a simulation study that goes beyond the established theory by examining the estimator's applicability in settings characterized by a generalized gamma distribution. We evaluate the effectiveness of the NPRIs and their estimates in modeling a real-world dataset related to gross domestic product per capita in the Americas.

Problem

• normalized scale-invariant indices
• gamma distribution
• unbiased estimation
• Gini coefficient
• variability indices

Innovation

• normalized scale-invariant indices
• unbiased estimation
• gamma distribution
• U-statistic
• Dirichlet proportions