Before revision:  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:  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.