We, in this paper, apply the smoothed and maximum empirical likelihood (EL) methods to construct the confidence intervals of the conditional quantile difference with left-truncated data. In particular, we prove the smoothed empirical log-likelihood ratio of the conditional quantile difference is asymptotically chi-squared when the observations with multivariate covariates form a stationary \(\alpha\)-mixing sequence. At the same time, we establish the asymptotic normality of the maximum EL estimator for the conditional quantile difference. A simulation study is conducted to investigate the finite sample behavior of the proposed methods and a real data application is provided.

在本文中，我们采用平滑和最大经验似然（EL）方法来构造带有左截断数据的条件分位数差的置信区间。特别地，当具有多元协变量的观测值形成平稳的\（\ alpha \）混合序列时，我们证明了条件分位数差异的平滑经验对数似然比是渐近卡方的。同时，我们为条件分位数差异建立了最大EL估计量的渐近正态性。进行了仿真研究，以研究所提出方法的有限样本行为，并提供了实际的数据应用程序。