The Advantages of Using Group Means in Estimating the Lorenz Curve and Gini Index From Grouped Data

A recent article proposed a histogram-based method for estimating the Lorenz curve and Gini index from grouped data that did not use the group means reported by government agencies. When comparing their method to one based on group means, the authors assume a uniform density in each grouping interval, which leads to an overestimate of the overall average income. After reviewing the additional information in the group means, it will be shown that as the number of groups increases, the bounds on the Gini index obtained from the group means become narrower. This is not necessarily true for the histogram method. Two simple interpolation methods using the group means are described and the accuracy of the estimated Gini index they yield and the histogram-based one are compared to the published Gini index for the 1967–2013 period. The average absolute errors of the estimated Gini index obtained from the two methods using group means are noticeably less than that of the histogram-based method. Supplementary materials for this article are available online. [Received August 2014. Revised September 2015.]

近期有学术文章提出了一种基于直方图的方法，可从分组数据中估计洛伦兹曲线（Lorenz curve）与基尼系数（Gini index），且无需使用政府机构公布的分组均值。在将该方法与基于分组均值的方法进行对比时，作者假设每个分组区间内服从均匀密度分布，这会导致对总体平均收入的高估。
在梳理分组均值中的附加信息后，本文将证明：随着分组数的增加，基于分组均值得到的基尼系数的置信界会逐渐收窄，但基于直方图的方法未必具备这一性质。
本文还介绍了两种基于分组均值的简单插值方法，并将这两种方法以及直方图法得到的基尼系数估计值，与1967—2013年期间已发布的基准基尼系数进行精度对比。结果显示，两种基于分组均值的方法得到的基尼系数估计值的平均绝对误差，显著低于直方图法的平均绝对误差。
本文的补充材料可在线获取。【2014年8月收稿，2015年9月修回】