How to (Not) Estimate Gini Coefficients for Fat Tailed Variables

Direct measurements of Gini coefficients by conventional arithmetic calculations are a poor estimator, even if paradoxically, they include the entire population, as because of super-additivity they cannot lend themselves to comparisons between units of different size, and intertemporal analyses are vitiated by the population changes. The Gini of aggregated units A and B will be higher than those of A and B computed separately. This effect becomes more acute with fatness of tails. When the sample size is smaller than entire population, the error is extremely high. The conventional literature on Gini coefficients cannot be trusted and comparing countries of different sizes makes no sense; nor does it make sense to make claims of "changes in inequality" based on conventional measures. We compare the standard methodologies to the indirect methods via maximum likelihood estimation of tail exponent. We compare to the tail method which is unbiased, with considerably lower error rate. We also consider measurement errors of the tail exponent and suggest a simple but efficient methodology to calculate Gini coefficients.