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Gini estimation under infinite variance

Author

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  • Andrea Fontanari
  • Nassim Nicholas Taleb
  • Pasquale Cirillo

Abstract

We study the problems related to the estimation of the Gini index in presence of a fat-tailed data generating process, i.e. one in the stable distribution class with finite mean but infinite variance (i.e. with tail index $\alpha\in(1,2)$). We show that, in such a case, the Gini coefficient cannot be reliably estimated using conventional nonparametric methods, because of a downward bias that emerges under fat tails. This has important implications for the ongoing discussion about economic inequality. 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-tailed distribution to that of a fat-tailed one, especially in the case of infinite variance. We also show how the nonparametric Gini bias increases with lower values of $\alpha$. We then prove that maximum likelihood estimation outperforms nonparametric methods, requiring a much smaller sample size to reach efficiency. Finally, for fat-tailed data, we provide a simple correction mechanism to the small sample bias of the nonparametric estimator based on the distance between the mode and the mean of its asymptotic distribution.

Suggested Citation

  • Andrea Fontanari & Nassim Nicholas Taleb & Pasquale Cirillo, 2017. "Gini estimation under infinite variance," Papers 1707.01370, arXiv.org, revised Dec 2017.
  • Handle: RePEc:arx:papers:1707.01370
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    References listed on IDEAS

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    1. Taleb, Nassim Nicholas & Douady, Raphael, 2015. "On the super-additivity and estimation biases of quantile contributions," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 429(C), pages 252-260.
    2. Li, Deli & Bhaskara Rao, M. & Tomkins, R. J., 2001. "The Law of the Iterated Logarithm and Central Limit Theorem for L-Statistics," Journal of Multivariate Analysis, Elsevier, vol. 78(2), pages 191-217, August.
    3. Eliazar, Iddo I. & Sokolov, Igor M., 2012. "Measuring statistical evenness: A panoramic overview," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 391(4), pages 1323-1353.
    4. Nolan, John P., 1998. "Parameterizations and modes of stable distributions," Statistics & Probability Letters, Elsevier, vol. 38(2), pages 187-195, June.
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    1. Mirco Nanni & Leandro Tortosa & José F Vicent & Gevorg Yeghikyan, 2020. "Ranking places in attributed temporal urban mobility networks," PLOS ONE, Public Library of Science, vol. 15(10), pages 1-25, October.

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