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.
Quantifying economic inequality from sample data is a central problem in applied economics and statistics. The classical Gini coefficient is arguably the most widely used rank-based inequality measure, but it is well known to exhibit non-negligible finitesample bias, especially for small and moderate sample sizes and for highly skewed populations; see, e.g., [2,5] and references therein. This motivates the development and analysis of alternative indices and estimators with improved small-sample properties, while retaining the desirable invariance and rank-based interpretation that make Ginitype measures attractive in practice.
A prominent extension of the Gini coefficient is the class of generalized and extended Gini indices, including the mth Gini index, S-Gini index and its extensions, which can be expressed as normalized contrasts of expected order statistics [5][6][7][12][13][14]. Such measures are particularly appealing because they admit transparent interpretations in terms of dispersion between extreme or intermediate ranks and connect naturally to the broader class of spectral (rank-dependent) inequality measures [3,4]. At the same time, many of these indices are typically estimated through U-statistic-type constructions that average rank contrasts over all subsamples of a given size m, often combined with normalization by the sample mean. Despite their conceptual appeal and practical relevance, the finite-sample bias behavior of these estimators has remained largely unexplored beyond specific cases.
The present paper contributes to this literature by developing a unified bias analysis for a broad family of order-statistic-based inequality indices and their canonical estimators. Specifically, for a non-negative random variable X with finite, strictly positive mean µ = E[X], and for m ⩾ 2, we consider the linear order-statistic inequality index
which includes as special cases the classical Gini coefficient (m = 2), the mth Gini index, extended mth Gini index, and S-Gini index (Table 1). This class also admits an integral representation I m = µ -1 1 0 w m (u) Q X (u)du, so that it can be viewed as a finite-dimensional approximation to continuous spectral measures of inequality.
Given an independent and identically distributed (i.i.d.) sample X 1 , . . . , X n with n ⩾ m, we study the estimator I m defined in (6), which averages the weighted orderstatistic contrasts over all m-subsamples and normalizes by X. Our first main result derives a general expression for the finite-sample bias Bias( I m , I m ) (Corollary 3). The bias can be decomposed into a linear combination of factors
which separates the contribution of each rank level in a manner that is amenable to analytic and numerical investigation. We further provide a characterization of ∆ n,r via Laplace-transform methods (Proposition 4), yielding a route to compute (or approximate) the bias under general non-negative distributions.
Our second main result establishes asymptotic unbiasedness: under mild moment conditions, Bias( I m , I m ) → 0 as n → ∞ (Proposition 5). While asymptotic unbiasedness is reassuring, it does not address the small-sample bias that often drives empirical discrepancies. The key contribution of the paper is therefore our third main result, which identifies a distributional setting where the estimator is exactly unbiased for all sample sizes. Specifically, we prove that if X ∼ Gamma(α, λ), then ∆ n,r = 0 for all r ⩽ n, implying (Corollary 3) Bias( I m , I m ) = 0 for all n ⩾ m, thereby extending earlier unbiasedness results for the Gini coefficient and its mth and extended variants [2,5,12,13]. The proof relies on the Dirichlet property of normalized gamma samples and the homogeneity of the maximum functional, which together yield the identity E[X r:r ] = µ E[X r:r /X].
Finally, we complement the theoretical analysis with a Monte Carlo study (Section 7) under gamma and non-gamma populations. The simulations confirm that the bias is essentially zero for Gamma(α, 1) distributions, even for n = 10, while non-gamma heavy-tailed alternatives such as Lognormal(0, 1) and Lomax(3, 1) display noticeable negative bias in small samples.
The rest of the paper unfolds as follows. Section 2 defines the index I m and establishes key properties and representations. Section 4 derives the general bias formula for I m and provides a Laplace-transform characterization. Section 5 proves asymptotic unbiasedness under moment conditions. Section 6 establishes exact unbiasedness under gamma populations. Section 7 reports Monte Carlo evidence, and Section 8 concludes.
Let X be a non-negative random variable with finite mean µ = E[X] > 0. For an i.i.d. random sample of size m ⩾ 2, denote the order statistics by
Define the population index
where (a 1 , . . . , a m ) are fixed real coefficients such that
The index I m measures a weighted contrast of expected rank positions relative to the population mean.
If the distribution of X is close to
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