  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.

Methodology