Semiparametric regression pursuit

The semiparametric partially linear model allows flexible modeling of covariate effects on the response variable in regression. It combines the flexibility of nonparametric regression and the parsimony of linear regression. The most important assumption in the existing methods for the estimation in this model is that a priori it is known which covariates have a linear effect and which do not. However, in applied work, this is rarely known in advance. We consider the problem of estimation in the partially linear models without assuming a priori which covariates have linear effects. We propose a semiparametric regression pursuit method for identifying the covariates with a linear effect. Our proposed method is a penalized regression approach using a group minimax concave penalty. Under suitable conditions we show that the proposed approach is model-pursuit consistent, meaning that it can correctly determine which covariates have a linear effect and which do not with high probability. The performance of the proposed method is evaluated using simulation studies that support our theoretical results. A data example is used to illustrated the application of the proposed method.