Robust exponential squared loss-based variable selection for high-dimensional single-index varying-coefficient model

Robust variable selection procedure through penalized regression has been gaining increased attention in the literature. They can be used to perform variable selection and are expected to yield robust estimates. In this article, we propose a robust variable selection procedure for high-dimensional single-index varying-coefficient model using penalized exponential squared loss. The proposed procedure simultaneously selects significant covariates with functional coefficients and local significant variables with parametric coefficients. With proper choices of penalty functions and regularization parameters, we show the asymptotic normality of the resulting estimate and further demonstrate that the proposed procedures perform as well as an oracle procedure. Our simulation studies reveal that our proposed method performs similarly to the oracle method in terms of the model error and the positive selection rate even in the presence of influential points. In contrast, other existing procedures have a much lower noncausal selection rate. Our analysis unravels the discrepancies of using our robust method versus the other penalized regression method, underscoring the importance of developing and applying robust penalized regression methods.