Paper: arXiv 2508.10663

Authors: Xia Han, Ruodu Wang, Qinyu Wu

Abstract

Via an axiomatic approach, we characterize the family of n-th order Gini deviation, defined as the expected range over n independent draws from a distribution, to quantify joint dispersion across multiple observations. This family extends the classical Gini deviation, which relies solely on pairwise comparisons. The normalized version is called a high-order Gini coefficient. The generalized indices grow increasingly sensitive to tail inequality as n increases, offering a more nuanced view of distributional extremes. The higher-order Gini deviations admit a Choquet integral representation, inheriting the desirable properties of coherent deviation measures. Furthermore, we show that both the n-th order Gini deviation and the n-th order Gini coefficient are statistically n-observation elicitable, allowing for direct computation through empirical risk minimization. Data analysis using World Inequality Database data reveals that higher-order Gini coefficients capture disparities that the classical Gini coefficient may fail to reflect, particularly in cases of extreme income or wealth concentration.

Complexity vs Empirical Score

  • Math Complexity: 8.0/10
  • Empirical Rigor: 7.5/10
  • Quadrant: Holy Grail — high math complexity, high empirical rigor

Why this score: This paper establishes a strong theoretical foundation for higher-order Gini indices using an axiomatic characterization, Choquet integral representations, and multi-observation elicitability. The theoretical framework is complemented by real-world empirical validation on wealth distributions using the World Inequality Database.

Research Flowchart

  flowchart TD
    A[Research Goal: Extend Classical Gini Index to Higher-Order n-th Order Joint Dispersion] --> B[Axiomatic Approach & Mathematical Characterization]
    B --> C[Derive Choquet Integral Representation & Prove Coherent Deviation Properties]
    B --> D[Establish n-Observation Elicitability for Empirical Risk Minimization]
    
    E[World Inequality Database WID Income/Wealth Data] --> F[Empirical Data Analysis & Higher-Order Gini Coefficient Computation]
    C --> F
    D --> F
    
    F --> G[Outcomes: Greater Sensitivity to Tail Inequality & Captures Disparities Missed by Classical Gini]