A Lorenz curve is a graphical representation of the distribution of income or wealth within a population. The generalized Lorenz curve can be created by scaling the values on the vertical axis of a Lorenz curve by the average output of the distribution. In this paper, we propose two non-parametric methods for testing the equality of two generalized Lorenz curves. Both methods are based on empirical likelihood and utilize a U-statistic. We derive the limiting distribution of the likelihood ratio, which is shown to follow a chi-squared distribution with one degree of freedom. We performed simulations to evaluate how well the proposed methods perform compared to an existing method, by examining their Type I error rates and power across different sample sizes and distribution assumptions. Our results show that the proposed methods exhibit superior performance in finite samples, particularly in small sample sizes, and are robust across various scenarios. Finally, we use real-world data to illustrate the methods of testing two generalized Lorenz curves.
翻译:洛伦兹曲线是表示一个人群内收入或财富分布的图形化工具。通过将分布的垂直轴上的值缩放为该分布的平均输出,可以创建广义洛伦兹曲线。在本文中,我们提出两种无参数方法来检验两个广义洛伦兹曲线的等价性。这两种方法都基于经验似然,并利用U统计量。我们推导出似然比的极限分布,其服从自由度为1的卡方分布。我们进行了模拟以评估所提出的方法在样本量和分布假设不同的情况下与现有方法相比的实验效果,通过检查它们的Ⅰ类错误率和功率。我们的结果表明,所提出的方法在有限样本中表现出超强的性能,特别是在小样本量的情况下,并且在各种情况下都具有鲁棒性。最后,我们使用实际数据来说明检验两个广义洛伦兹曲线的方法。