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. Second, we show how the maximum likelihood estimator outperforms the non-parametric requiring a much smaller sample size to reach efficiency.
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-tailed distribution to that of a fat-tailed one, especially in the case of infinite variance. We show how the maximum likelihood estimator outperforms the non-parametric requiring a much smaller sample size to reach efficiency.