The Lorenz curve portrays the inequality of income distribution. In this article, we develop three modified empirical likelihood (EL) approaches including adjusted empirical likelihood, transformed empirical likelihood, and transformed adjusted empirical likelihood to construct confidence intervals for the generalized Lorenz ordinate. We have shown that the limiting distribution of the modified EL ratio statistics for the generalized Lorenz ordinate follows the scaled Chi-Squared distributions with one degree of freedom. The coverage probabilities and mean lengths of confidence intervals are compared of the proposed methods with the traditional EL method through simulations under various scenarios. Finally, the proposed methods are illustrated using a real data application to construct confidence intervals.

翻译：洛伦兹曲线刻画了收入分配的不平等程度。本文提出了三种修正经验似然（EL）方法，包括调整后经验似然、变换后经验似然以及变换后调整经验似然，用于构建广义洛伦兹坐标的置信区间。我们证明了修正EL比率统计量关于广义洛伦兹坐标的极限分布服从自由度为1的尺度卡方分布。通过多种场景下的模拟研究，将所提方法的覆盖概率和置信区间平均长度与传统EL方法进行了比较。最后，通过实际数据应用展示了所提方法在构建置信区间方面的效果。