Exact finite-sample bias, variance, and MSE of normalized scale-invariant inequality indices under Gamma mixture models
Abstract
We study the finite-sample properties of estimators of a broad class of normalized scale-invariant inequality indices (NSIIs) under finite Gamma mixture models. The class includes the Gini coefficient, the mth Gini index, the Extended mth Gini index, linear order-statistic inequality indices, and several entropy- and dispersion-based measures. Building on the Gamma--Dirichlet representation, we derive exact expressions for the expectation and finite-sample bias of the natural estimator of an NSII based on a U-statistic when the population follows a finite Gamma mixture with a common rate parameter. The resulting bias formula explicitly characterizes the effects of the mixture weights and component shape parameters. We also establish that the bias converges to zero as the sample size increases, yielding asymptotic unbiasedness. Furthermore, we derive exact finite-sample expressions for the variance and mean squared error of the estimator by accounting for the overlap structure among the subsets entering the U-statistic. In addition, we establish strong consistency and asymptotic normality. When the mixture degenerates to a single Gamma distribution, the finite-sample bias vanishes, recovering the exact unbiasedness result previously obtained for NSIIs under the Gamma model. Numerical illustrations and simulation studies are provided to investigate the finite-sample behavior of the estimator and to illustrate the theoretical findings. Overall, the proposed framework provides a unified characterization of finite-sample estimation under heterogeneous Gamma populations and quantifies the effects of mixture heterogeneity on bias, variance, and mean squared error.
Related
- Unbiased estimation of normalized scale-invariant indices under the Gamma distribution
- Sandwich Variance Estimation in Scale Mixture of Skew-Normal Linear Mixed Models.
- Faber--Krahn inequality for homogeneous normalized p-Laplacian type operators
- A logarithmic Brascamp--Lieb variance inequality
- Variance lower bounds for geometric functionals of rotationally invariant log-concave random polytopes
BibTeX
@misc{vila2026exact,
title = {{Exact finite-sample bias, variance, and MSE of normalized scale-invariant inequality indices under Gamma mixture models}},
author = {Roberto Vila and Felipe Quintino},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.05621},
eprint = {2610.05621},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.05621},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Exact finite-sample bias, variance, and MSE of normalized scale-invariant inequality indices under Gamma mixture models
AU - Roberto Vila
AU - Felipe Quintino
PY - 2026
DA - 2026/10/04
PB - arXiv
DO - 10.48550/arxiv.2610.05621
UR - https://arxiv.org/abs/2610.05621
AB - We study the finite-sample properties of estimators of a broad class of normalized scale-invariant inequality indices (NSIIs) under finite Gamma mixture models. The class includes the Gini coefficient, the mth Gini index, the Extended mth Gini index, linear order-statistic inequality indices, and several entropy- and dispersion-based measures. Building on the Gamma--Dirichlet representation, we derive exact expressions for the expectation and finite-sample bias of the natural estimator of an NSII based on a U-statistic when the population follows a finite Gamma mixture with a common rate parameter. The resulting bias formula explicitly characterizes the effects of the mixture weights and component shape parameters. We also establish that the bias converges to zero as the sample size increases, yielding asymptotic unbiasedness. Furthermore, we derive exact finite-sample expressions for the variance and mean squared error of the estimator by accounting for the overlap structure among the subsets entering the U-statistic. In addition, we establish strong consistency and asymptotic normality. When the mixture degenerates to a single Gamma distribution, the finite-sample bias vanishes, recovering the exact unbiasedness result previously obtained for NSIIs under the Gamma model. Numerical illustrations and simulation studies are provided to investigate the finite-sample behavior of the estimator and to illustrate the theoretical findings. Overall, the proposed framework provides a unified characterization of finite-sample estimation under heterogeneous Gamma populations and quantifies the effects of mixture heterogeneity on bias, variance, and mean squared error.
ER -