Before revision:  While the Gini index is a measurement of fat tailedness, little attention has been brought to a significant downward bias in conventional applications, one that increases with lower values of \alpha. First, we show how the "non-parametric" 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 tail distribution to the domain of attraction of a fat tailed, infinite variance one.
After revision:  We show that, in such a case, the Gini coefficient can not be reliably estimated using conventional nonparametric methods, because of a downward bias that emerges in case of fat tails. 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 tail distribution to the domain of attraction of a fat tailed, infinite variance one.