We consider a linear model which can have a large number of explanatory variables, the errors with an asymmetric distribution or some values of the explained variable are missing at random. In order to take in account these several situations, we consider the non parametric empirical likelihood (EL) estimation method. Because a constraint in EL contains an indicator function then a smoothed function instead of the indicator will be considered. Two smoothed expectile maximum EL methods are proposed, one of which will automatically select the explanatory variables. For each of the methods we obtain the convergence rate of the estimators and their asymptotic normality. The smoothed expectile empirical log-likelihood ratio process follow asymptotically a chi-square distribution and moreover the adaptive LASSO smoothed expectile maximum EL estimator satisfies the sparsity property which guarantees the automatic selection of zero model coefficients. In order to implement these methods, we propose four algorithms.

翻译：我们考虑一类线性模型，该模型可能包含大量解释变量，误差项呈非对称分布，或解释变量的某些值随机缺失。为应对这些多重情形，我们采用非参数经验似然估计方法。由于经验似然中的约束条件包含示性函数，本文将采用平滑函数替代示性函数。本文提出两种平滑期望分位数最大经验似然方法，其中一种能够自动选择解释变量。针对每种方法，我们获得了估计量的收敛速度及其渐近正态性。平滑期望分位数经验对数似然比过程渐近服从卡方分布，且自适应LASSO平滑期望分位数最大经验似然估计量满足稀疏性性质，保证了模型零系数的自动选择。为实现这些方法，我们提出了四种算法。