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方法（包括调整经验似然、变换经验似然及变换调整经验似然），用于构建广义洛伦兹纵坐标的置信区间。我们证明，广义洛伦兹纵坐标的修正经验似然比统计量的极限分布服从自由度为1的缩放卡方分布。通过多场景模拟，对比了所提方法与传统经验似然方法在置信区间覆盖概率和平均长度方面的表现。最后，利用真实数据应用案例展示了所提方法在置信区间构建中的有效性。