Unbiased estimation of normalized scale-invariant indices under the Gamma distribution
Abstract
We introduce a broad class of normalized scale-invariant indices (NSIIs) generated by measurable symmetric 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 NSII in the proposed class 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 evaluate the effectiveness of the NSIIs and their estimates in modeling a real-world dataset related to gross domestic product per capita in the Americas.
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BibTeX
@misc{vila2026unbiased,
title = {{Unbiased estimation of normalized scale-invariant indices under the Gamma distribution}},
author = {Roberto Vila and Helton Saulo and Felipe Quintino},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2606.22712},
eprint = {2606.22712},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2606.22712},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Unbiased estimation of normalized scale-invariant indices under the Gamma distribution
AU - Roberto Vila
AU - Helton Saulo
AU - Felipe Quintino
PY - 2026
DA - 2026/10/03
PB - arXiv
DO - 10.48550/arxiv.2606.22712
UR - https://arxiv.org/abs/2606.22712
AB - We introduce a broad class of normalized scale-invariant indices (NSIIs) generated by measurable symmetric 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 NSII in the proposed class 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 evaluate the effectiveness of the NSIIs and their estimates in modeling a real-world dataset related to gross domestic product per capita in the Americas.
ER -