Structure identification for partially linear partially concave models

Partially linear partially concave models are semiparametric regression models that can capture linear and concavity-constrained nonlinear effects within one framework. A fundamental problem of this kind of model is deciding which covariates have linear effects and which covariates have strictly concave effects. Assuming that the true regression function is partially linear partially concave and sparse, we develop two structure selection procedures for classifying the covariates into linear, strictly concave, and irrelevant subsets. We show that the procedures based on penalized concavity-constrained additive regressions can correctly identify structures even if the underlying true functions are nonadditive; namely, the proposed procedures are additively faithful in a general setting. We prove that consistent structure selection is achievable when the total number of covariates and the number of concave covariates grow at polynomial rates with sample size. We introduce algorithms to implement the proposed procedures and demonstrate their performance by simulation analysis.