Robust variable selection for partially linear additive models

Abstract

Among semiparametric regression models, partially linear additive models provide a useful tool to include additive nonparametriccomponents as well as a parametric component, when explaining the relationship between the response and a set of explanatory variables. This paper concerns such models under sparsity assumptions for the covariates included in the linear component. Sparse statistical models are easier to interpret than dense ones, since only a small number of the parameters are non--zero. This scenario is common in regression problems, making variable selection an important task. As in other settings, outliers either in the residuals or in the covariates involved in the linear component have a harmful effect. To simultaneously achieve model selection for the parametric component of the model and resistance to outliers, we combine preliminary robust estimators of the additive component, robust linear $MM-$regression estimators with a penalty such as SCAD on the coefficients in the parametric part. Under mild assumptions, consistency results and rates of convergence for the proposed estimators are derived. A Monte Carlo study is carried out to compare, under different models and contamination schemes, the performance of the robust proposal with its classical counterpart. The numerical results show the advantage of using the robust approach. Through the analysis of a real data set, we also illustrate the benefits of the proposed procedure.