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 $m$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.