Bias analysis of a linear order-statistic inequality index estimator: Unbiasedness under gamma populations

TL;DR

This work develops a unified bias analysis for a broad class of rank-based inequality measures defined by linear combinations of expected order statistics. It introduces a general population index $I_m$ and a U-statistic-type estimator $\widehat{I}_m$, deriving a general bias decomposition via $\Delta_{n,r}=\mathbb{E}[X_{r:r}/\overline{X}] - \mathbb{E}[X_{r:r}]/\mu$ and a Laplace-transform characterization to compute the bias under broad distributions. Under mild moment conditions, it proves asymptotic unbiasedness $\mathrm{Bias}(\widehat{I}_m,I_m)\to 0$, and shows exact unbiasedness for Gamma populations since $\Delta_{n,r}=0$ for all $n\ge r$, extending known results for Gini-type estimators. Monte Carlo simulations corroborate the gamma-unbiasedness and reveal non-negligible small-sample bias for heavy-tailed non-gamma distributions, highlighting the practical impact of distributional choice on finite-sample performance.

Abstract

This paper studies a class of rank-based inequality measures built from linear combinations of expected order statistics. The proposed framework unifies several well-known indices, including the classical Gini coefficient, the $m$th Gini index, extended $m$th Gini index and $S$-Gini index, and also connects to spectral inequality measures through an integral representation. We investigate the finite-sample behavior of a natural U-statistic-type estimator that averages weighted order-statistic contrasts over all subsamples of fixed size and normalizes by the sample mean. A general bias decomposition is derived in terms of components that isolate the effect of random normalization on each rank level, yielding analytical expressions that can be evaluated under broad non-negative distributions via Laplace-transform methods. Under mild moment conditions, the estimator is shown to be asymptotically unbiased. Moreover, we prove exact unbiasedness under gamma populations for any sample size, extending earlier unbiasedness results for Gini-type estimators. A Monte Carlo study is performed to numerically check that the theoretical unbiasednes under gamma populations.

Theorems & Definitions (11)

• Remark 1
• Proposition 1
• proof
• Theorem 2
• proof
• Corollary 3
• Proposition 4
• proof
• Proposition 5
• proof