英文标题：
《Gini estimation under infinite variance》
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作者：
Andrea Fontanari, Nassim Nicholas Taleb, Pasquale Cirillo
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最新提交年份：
2017
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英文摘要：
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.
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中文摘要：
我们研究了存在厚尾数据生成过程的基尼指数估计相关问题，即具有有限平均值但无限方差的稳定分布类中的数据生成过程（即具有尾指数$\\α\\ in（1,2）$）。我们表明，在这种情况下，基尼系数无法使用传统的非参数方法可靠估计，因为在厚尾下会出现向下的偏差。这对正在进行的关于经济不平等的讨论具有重要影响。我们首先讨论基尼指数的非参数估计如何在其渐近分布的对称结构中经历相变，因为数据分布从轻尾分布的吸引域转移到厚尾分布的吸引域，特别是在无穷方差的情况下。我们还展示了非参数基尼偏差如何随着$\\α$值的降低而增加。然后，我们证明了最大似然估计优于非参数方法，需要更小的样本量才能达到效率。最后，对于厚尾数据，我们基于模式与其渐近分布均值之间的距离，为非参数估计的小样本偏差提供了一种简单的校正机制。
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分类信息：

一级分类：Statistics 统计学
二级分类：Methodology 方法论
分类描述：Design, Surveys, Model Selection, Multiple Testing, Multivariate Methods, Signal and Image Processing, Time Series, Smoothing, Spatial Statistics, Survival Analysis, Nonparametric and Semiparametric Methods
设计，调查，模型选择，多重检验，多元方法，信号和图像处理，时间序列，平滑，空间统计，生存分析，非参数和半参数方法
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一级分类：Quantitative Finance 数量金融学
二级分类：Risk Management 风险管理
分类描述：Measurement and management of financial risks in trading, banking, insurance, corporate and other applications
衡量和管理贸易、银行、保险、企业和其他应用中的金融风险
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