Abstract The Pietra ratio (Pietra index) is also known as the Robin Hood index or Schutz coefficient (Ricci–Schutz index). It is a measure of statistical heterogeneity in positive random variables. In this paper, we propose the jackknife empirical likelihood (JEL), the adjusted JEL, the extended JEL, and the balanced adjusted JEL method, for interval estimation of the Pietra ratio. We compare the performance of the proposed methods with the normal approximation (NA), bootstrap based methods and NA jackknife method. Simulation results indicate that under both symmetric and skewed distributions, the extended JEL method gives the best performance in terms of coverage probability. We illustrate the proposed methods by applying our methods to investigate the income data from the 2013 Current Population Survey conducted by the US Census Bureau.

中文翻译：

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Pietra 比率的 Jackknife 经验似然推断

摘要 彼得拉比（Pietra 指数）也称为罗宾汉指数或舒茨系数（Ricci-Schutz 指数）。它是对正随机变量中统计异质性的度量。在本文中，我们提出了 jackknife 经验似然 (JEL)、调整 JEL、扩展 JEL 和平衡调整 JEL 方法，用于 Pietra 比率的区间估计。我们将所提出的方法的性能与正态近似（NA）、基于引导的方法和 NA jackknife 方法的性能进行了比较。仿真结果表明，在对称分布和偏态分布下，扩展 JEL 方法在覆盖概率方面给出了最佳性能。我们通过应用我们的方法来调查美国人口普查局进行的 2013 年当前人口调查的收入数据来说明所提出的方法。