Title: Gini estimation under infinite variance,
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.