Variable Selection Based on Profile Forward Selection of Partial Linear Models with Interactive Terms

Due to the rapid development of information technology and data acquisition technology, the model which only considers the linear main effect can not provide accurate prediction results, and the interaction between the predictor and response variables can not be ignored, so the variable selection problem of the model with interaction terms has become an important research topic in the statistical analysis today. In this paper, we discuss the problem of variable selection for a partially linear model with interaction terms using the profile forward selection method under high dimensional data. We propose the two-stage interactive selection algorithm (iPFST) under strong genetic condition and the profile forward selection algorithm (iPFSM) under marginality principle respectively. Theoretically, we use the consistency of profile estimators to prove that profile estimators have uniform convergence rate, and use the screening consistency to prove that iPFST algorithm and iPFSM algorithm can uniformly identify all important linear main effect terms and important interaction effect terms with probability 1. Seven regularization conditions for the theorem are given. Numerical simulation shows the superiority of iPFST and iPFSM in variable selection, and the two algorithms are compared, then iPFST algorithm is better than iPFSM algorithm. Finally, we give detailed technical proof.