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：本文中，洛伦兹曲线描绘了收入分配的不平等现象。本文提出了三种修正经验似然方法，包括调整经验似然、转换经验似然和转换调整经验似然，以构建广义洛伦兹纵坐标的置信区间。我们证明，对于广义洛伦兹纵坐标的修正EL比统计量的极限分布是自由度为一的缩放卡方分布。通过各种场景的模拟，我们比较了所提出的方法与传统EL方法的置信区间覆盖概率和平均长度。最后，我们使用实际数据应用说明所提出的方法以构建置信区间。