The Study For Variable Selection In Partially Linear Model

The partially linear model is a widely used class of semi-parametric model,which was ?rstly put forward by Engle(1986).Because the model contains both the parameter part and the non-parametric part,it has good interpretative of parametric regression model and ?exibility of non-parametric regression models,which is widely used for data ?tting processing in practice.However, when we operate the data that need to ?t,we often have to be faced with processing high-dimensional data.But we know that high-dimensional data with internal sparse, ie with dimension p increasing,the proportion of the number of data points that contained in a local neighborhood is more and more small throughout the sample.It will lead to estimation and ?tting accuracy decreasing rapidly and resulted in so-called ”curse of dimensionality”,which is derived variable selection problem. Variable selection has become one of the hot topics of statistical research,it has become a fundamental method to improve prediction precision and interpretability of model. It has the important practical signi?cance and research value.In this paper,we consider variable selection and model estimation in partially linear model under the assumption that the vector of regression coe?cients is sparse.We mainly discuss using two types of penalty methods in variable selection problems, one use Adaptive Lasso method that penalize linear part,under the condition of assuming that there is no regression relationship between parameter part and the non-parametric part, we prove that the penalized estimator has the Oracle property,in the sense that the existed penalized doesn’t e?ect the estimation of non-zero coe?cients, and it can accurately identify the non-zero coe?cient. Another method use the double penalties to penalize both the parameter and the non-parametric part at the same time, we also discuss the penalized estimator has the Oracle property,and it can accurately identify the non-zero coe?cient. Numerical simulation study is conducted to demonstrate the e?ective performance in the ?nite sample.