Robust Variable Selection for Single-Index Varying-Coefficient Model with Missing Data in Covariates

As applied sciences grow by leaps and bounds, semiparametric regression analyses have broad applications in various fields, such as engineering, finance, medicine, and public health. Single-index varying-coefficient model is a common class of semiparametric models due to its flexibility and ease of interpretation. The standard single-index varying-coefficient regression models consist mainly of parametric regression and semiparametric regression, which assume that all covariates can be observed. The assumptions are relaxed by taking the models with missing covariates into consideration. To eliminate the possibility of bias due to missing data, we propose a probability weighted objective function. In this paper, we investigate the robust variable selection for a single-index varying-coefficient model with missing covariates. Using parametric and nonparametric estimates of the likelihood of observations with fully observed covariates, we examine the estimators for estimating the likelihood of observations. For variable selection, we use a weighted objective function penalized by a non-convex SCAD. Theoretical challenges include the treatment of missing data and a single-index varying-coefficient model that uses both the non-smooth loss function and the non-convex penalty function. We provide Monte Carlo simulations to evaluate the performance of our approach.