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Gini estimation under infinite variance

Andrea Fontanari, Nassim Nicholas Taleb, Pasquale Cirillo

posted on 12 July 2017

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Under infinite variance, the Gini coefficient cannot be reliably estimated using conventional "non-parametric" methods.
We study different approaches to the estimation of the Gini index in presence of a heavy tailed data generating process, that is, one with Paretan tails and/or in the stable distribution class with finite mean but non-finite variance (with tail index $\alpha\in(1,2)$).
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.
Second, we show how the maximum likelihood estimator outperforms the non-parametric requiring a much smaller sample size to reach efficiency.
Finally we provide a simple correction mechanism to the small sample bias of the non-parametric estimator based on the distance between the mode and the mean of its asymptotic distribution for the case of heavy tailed data generating process.