Under infinite variance, the Gini coefficient cannot be reliably estimated using conventional nonparametric 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-paramteric" estimator based on the distance between the mode and the mean of its asymptotic distribution for the case of heavy tailed data generating process.
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.
R
nonparametric
"non-parametric"
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D
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R
"non-paramteric"
non-parametric