Higher-order Gini indices: An axiomatic approach

TL;DR

This paper introduces higher-order Gini indices, extending classical Gini measures to better capture distributional extremes and tail inequality through an axiomatic, coherent, and statistically elicitable framework.

Contribution

It characterizes the family of n-th order Gini deviations and coefficients, extending classical measures with a new axiomatic foundation and demonstrating their statistical and practical advantages.

Findings

Higher-order Gini coefficients are more sensitive to tail inequality.

They can be directly computed via empirical risk minimization.

Data analysis shows they reveal disparities missed by classical Gini.

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.