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.
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