Robust check loss-based variable selection of high-dimensional single-index varying-coefficient model

• Develop a variable selection method for high-dimensional single-index varying-coefficient models using a shrinkage idea.
• Simultaneously select significant covariates with functional coefficients and local significant variables with parametric coefficients.
• Under defined regularity conditions, with appropriate selection of tuning parameters, the consistency of the variable selection procedure and the oracle property of the estimators are established. The method is illustrated with numerical simulations.
• Due to the robustness of the check loss function to outliers in the finite samples, our proposed variable selection method is more robust than the ones based on the least squares criterion.
• Propose to use the Difference Convex algorithm to solve the corresponding non-convex optimization problem.

Single-index varying-coefficient model is an important mathematical modeling method to model nonlinear phenomena in science and engineering. In this paper, we develop a variable selection method for high-dimensional single-index varying-coefficient models using a shrinkage idea. The proposed procedure can simultaneously select significant nonparametric components and parametric components. Under defined regularity conditions, with appropriate selection of tuning parameters, the consistency of the variable selection procedure and the oracle property of the estimators are established. Moreover, due to the robustness of the check loss function to outliers in the finite samples, our proposed variable selection method is more robust than the ones based on the least squares criterion. Finally, the method is illustrated with numerical simulations.