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：洛伦兹曲线是描述人口中收入或财富分布的一种图形表示方法。广义洛伦兹曲线可通过将洛伦兹曲线纵轴数值乘以分布的平均产出得到。本文提出两种检验广义洛伦兹曲线相等性的非参数方法，两种方法均基于经验似然并利用U统计量。我们推导了似然比的极限分布，证明其服从自由度为1的卡方分布。通过模拟实验，我们考察了不同样本量和分布假设下所提方法与现有方法的I类错误率和检验功效，评估其性能表现。结果表明，所提方法在有限样本中展现出优越性能，尤其在小样本情况下表现突出，且在不同场景下具有稳健性。最后，我们运用真实数据实例演示了两种广义洛伦兹曲线检验方法的应用。