Gamma混合模型下归一化尺度不变不平等指数的精确有限样本偏差、方差和均方误差
Exact finite-sample bias, variance, and MSE of normalized scale-invariant inequality indices under Gamma mixture models
- Department of Statistics, University of Brasilia（巴西利亚大学统计系）
- University of Brasilia（巴西利亚大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本研究针对Gamma混合模型下的归一化尺度不变不平等指数，推导了估计量的精确有限样本偏差、方差和均方误差公式，并证明其渐近无偏性、强相合性和渐近正态性，为异质总体下的估计提供了统一框架。
AI中文摘要：
我们研究了有限Gamma混合模型下一大类归一化尺度不变不平等指数（NSIIs）估计量的有限样本性质。该类别包括基尼系数、第m个基尼指数、扩展第m个基尼指数、线性次序统计量不平等指数，以及若干基于熵和离散度的度量。基于Gamma-Dirichlet表示，当总体服从具有共同速率参数的有限Gamma混合时，我们推导了基于U统计量的NSII自然估计量的期望和有限样本偏差的精确表达式。所得的偏差公式明确刻画了混合权重和分量形状参数的影响。我们还证明了随着样本量增加偏差收敛于零，从而得到渐近无偏性。此外，通过考虑进入U统计量的子集之间的重叠结构，我们推导了估计量方差和均方误差的精确有限样本表达式。另外，我们建立了强相合性和渐近正态性。当混合退化为单一Gamma分布时，有限样本偏差消失，恢复了先前在Gamma模型下针对NSIIs获得的精确无偏性结果。我们提供了数值示例和模拟研究，以考察估计量的有限样本行为并说明理论发现。总体而言，所提出的框架为异质Gamma总体下的有限样本估计提供了统一刻画，并量化了混合异质性对偏差、方差和均方误差的影响。
英文摘要：
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 $m$th Gini index, the Extended $m$th 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.
