  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 and estimating equations 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. A first
inspection on the robustness of estimating equations contributing to the
estimating equations selection and variable selection is discussed from both
theoretical perspective and intuitive simulation results in this paper. 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 robustness of the estimator is measured via the bounded influence function,
and 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 the robustness of sparse estimating equations selection fundamentally
enhances variable selection accuracy when the data have heavy tails and/or
include underlying outliers.
