In this paper, we study penalized empirical likelihood for parameter estimation and variable selection in partially linear models with measurement errors in possibly all the variables. By using adaptive Lasso penalty function, we show that penalized empirical likelihood has the oracle property. That is, with probability tending to one, penalized empirical likelihood identifies the true model and estimates the nonzero coefficients as efficiently as if the sparsity of the true model was known in advance. Also, we introduce the penalized empirical likelihood ratio statistic to test a linear hypothesis of the parameter and prove that it follows an asymptotic Chi-square distribution under the null hypothesis. Some simulations and an application are given to illustrate the performance of the proposed method.

在本文中，我们研究了部分线性模型中可能存在所有变量的测量误差的参数估计和变量选择的惩罚性经验似然。通过使用自适应拉索惩罚函数，我们证明了惩罚性经验似然具有预言性。即，随着概率趋向于一，受惩罚的经验似然识别真实模型并有效地估计非零系数，就好像预先知道真实模型的稀疏性一样。此外，我们引入了惩罚性经验似然比统计量以检验参数的线性假设，并证明其在零假设下遵循渐近卡方分布。给出了一些仿真和应用实例，说明了该方法的性能。