Gini Estimation Under Infinite Variance

22 Pages Posted: 21 Jul 2017 Last revised: 22 Dec 2017

See all articles by Andrea Fontanari

Andrea Fontanari

Delft University of Technology - Delft Institute of Applied Mathematics (DIAM)

Nassim Nicholas Taleb

New York University (NYU) - NYU Tandon School of Engineering

Pasquale Cirillo

Delft University of Technology; Delft University of Technology - Delft Institute of Applied Mathematics (DIAM)

Date Written: July 19, 2017

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 α ∈ (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 in case of 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 show how the nonparametric Gini bias increases with lower values of α. 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.

Keywords: Gini index; inequality measure; size distribution; extremes; α-stable distribution

JEL Classification: D31; C46

Suggested Citation

Fontanari, Andrea and Taleb, Nassim Nicholas and Cirillo, Pasquale, Gini Estimation Under Infinite Variance (July 19, 2017). Available at SSRN: https://ssrn.com/abstract=3005184 or http://dx.doi.org/10.2139/ssrn.3005184

Andrea Fontanari

Delft University of Technology - Delft Institute of Applied Mathematics (DIAM) ( email )

Mekelweg 4
Delft, Holland 2628
Netherlands

Nassim Nicholas Taleb

New York University (NYU) - NYU Tandon School of Engineering ( email )

6 MetroTech Center
Brooklyn, NY 11201
United States

Pasquale Cirillo (Contact Author)

Delft University of Technology ( email )

Stevinweg 1
Stevinweg 1
Delft, 2628 CN
Netherlands

Delft University of Technology - Delft Institute of Applied Mathematics (DIAM) ( email )

Mekelweg 4
Delft, Holland 2628
Netherlands

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