  As an effective nonparametric method, empirical likelihood (EL) is appealing
in combining estimating equations flexibly and adaptively for incorporating
data information. To select important variables in the sparse high-dimensional
model, we consider a penalized EL method based on robust estimating functions
by applying two penalty functions for regularizing the regression parameters
and the associated Lagrange multipliers simultaneously, which allows the
dimensionalities of both regression parameters and estimating equations to grow
exponentially with the sample size. The proposed method can improve the
robustness and effectiveness when the data have underlying outliers or heavy
tails in the response variables and/or covariates. The oracle properties are
also established under some regularity conditions. Extensive simulation studies
and a yeast cell data are used to evaluate the performance of the proposed
method. The numerical results reveal that robust remedies on estimating
equations are needed when the data have heavy tails and/or include underlying
outliers.
