Gini estimation under infinite variance

TL;DR

This paper investigates the challenges of estimating the Gini index in fat-tailed distributions with infinite variance, revealing biases in conventional methods and proposing a maximum likelihood approach with bias correction.

Contribution

It demonstrates the limitations of nonparametric Gini estimation under infinite variance and introduces a more efficient maximum likelihood method with a bias correction mechanism.

Findings

Nonparametric Gini estimator exhibits downward bias in fat-tailed data.

Maximum likelihood estimation outperforms nonparametric methods in such contexts.

A simple bias correction based on distribution mode and mean improves estimates.

Abstract

We study the problems related to the estimation of the Gini index in presence of a fat-tailed data generating process, i.e. one in the stable distribution class with finite mean but infinite variance (i.e. with tail index $\alpha\in(1,2)$). We show that, in such a case, the Gini coefficient cannot be reliably estimated using conventional nonparametric methods, because of a downward bias that emerges under fat tails. This has important implications for the ongoing discussion about economic inequality. We start by discussing how the nonparametric estimator of the Gini index undergoes a phase transition in the symmetry structure of its asymptotic distribution, as the data distribution shifts from the domain of attraction of a light-tailed distribution to that of a fat-tailed one, especially in the case of infinite variance. We also show how the nonparametric Gini bias increases with lower values of $\alpha$. We then prove that maximum likelihood estimation outperforms nonparametric methods, requiring a much smaller sample size to reach efficiency. Finally, for fat-tailed data, we provide a simple correction mechanism to the small sample bias of the nonparametric estimator based on the distance between the mode and the mean of its asymptotic distribution.