The Interpolation of the Lorenz Curve and Gini Index from Grouped Data
基于分组数据的洛伦兹曲线与基尼指数插值AI 解读 ↓
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基于分组数据的洛伦兹曲线与基尼指数插值
- 研究问题
- 仅有分组收入数据时，如何插值得到平滑的洛伦兹曲线并估计基尼指数，同时检查曲线是否满足凸性与单调性要求？
- 方法与思路
- 作者将数值分析中的分段 Hermite 插值用于洛伦兹曲线，讨论插值曲线的凸性或单调性条件，并考察基尼指数估计。
- 材料陈述的贡献
- 摘要称，该插值方法通常能得到准确的基尼指数估计，并给出凸性或单调性条件，同时用理论例子展示可能出现的错误。
- 阅读关注点
- 重点核对插值需要哪些分组信息、凸性与单调性条件如何检验，以及所得基尼指数估计与既有上下界是否相容。
- 阅读边界
- 摘要明确指出，分段 Hermite 插值不总能保持洛伦兹曲线应有的凸性；未提供具体误差结果，不能据此认定方法普遍有效。
原文摘要
ECONOMISTS OFTEN SUMMARIZE the income distribution by the Lorenz curve and Gini index. A variety of parametric methods (e.g., [1 and 8]) have been developed to estimate these measures from the grouped income data governments provide (e.g., [3 and 12]). Previously, one of the authors developed a distribution-free approach [5] which yielded accurate bounds on the Gini index. While analogous bounds on the Lorenz curve can be obtained [5 and 10], the resulting curve is not smooth so a method of interpolation is needed. The purpose of this paper is to adapt an old technique of numerical analysis, Hermite interpolation [7 and 13], to our problem and to show that it usually works in theory and in practice. Our paper was motivated by the work of Brittain [2] who also used numerical methods. Unfortunately, his procedure often resulted in estimates of the Gini index which were inconsistent with the above-mentioned bounds. Although the piecewise Hermite interpolation yielded accurate estimates of the Gini index, it is not always convex as the Lorenz curve must be. Section 5 states conditions for the interpolated curve to be convex or at least increasing over an interval. While these conditions are usually satisfied by real data, a theoretical example illustrates how an error may arise.