Beware the Gini Index! A New Inequality Measure
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The Gini index underestimates inequality for heavy-tailed distributions: for example, a Pareto distribution with exponent 1.5 (which has infinite variance) has the same Gini index as any exponential distribution (a mere 0.5). This is because the Gini index is relatively robust to extreme observations: while a statistic's robustness to extremes is desirable for data potentially distorted by outliers, it is misleading for heavy-tailed distributions, which inherently exhibit extremes. We propose an alternative inequality index: the variance normalized by the second moment. This ratio is more stable (hence more reliable) for large samples from an infinite-variance distribution than the Gini index paradoxically. Moreover, the new index satisfies the normative axioms of inequality measurement; notably, it is decomposable into inequality within and between subgroups, unlike the Gini index.
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