Variable Selection For Ultrahigh Dimensional Partially Linear Models

As one of the most commonly used semiparametric statistical models,the partially linear model not only retains the characteristics of linear regression model that it is intuitive and easy to explain,but also partially inherits the robustness of nonparametric regression model.The par-tially linear model is a perfect combination of linear model and nonparametric regression model,and it has attracted the attention of many statisticians such that a large number of research results are emerged.These statisticians developed many statistical methods by combining the estima-tion methods of the parametric and nonparametric regression model,and they derived the large sample properties of the estimators for the parametric components of interest and nonparametric component under some regularity conditions.These research results have opened up an impor-tant field of statistics.At the same time,the partially linear model has become a crucial tool for dealing with data in a large variety of areas,such as economics,finance and biomedicine,etc.However,with the rapid development of information technology and computer,the accu-mulation of data is increasing,and the range of data is more and more wide.People gradually recognize the importance of high-dimensional data and carry out in-depth research.At present,a lot of achievements have been made for variable selection in the high-dimensional linear mod-el.However,the literature about the high-dimensional partially linear model is relatively few,especially for ultrahigh dimensional data.Based on profile least squares and regularization after retention(RAR)method,this paper proposes a new method to perform variable selection for ultrahigh dimensional partially linear model.Firstly,we transform the model into the ultrahigh dimensional linear model by using the profile technique.Secondly,we retain some important variables according to the marginal correlation coefficient,and conduct Lasso penalty only for the remaining variables.Thus we can obtain the partially penalized least square estimator of parameter vector in the linear part.Fi-nally,we provide a relevant algorithm to realize the proposed method.In this paper,the penalty method is used for ultrahigh dimensional partially linear model,which broadens the idea for the study of ultrahigh dimensional data.In addition,the partially penalty method is used to improve the accuracy of the estimator,and it is easy to realized by programming.Under certain regular-ity conditions,it is proved that the estimator achieves sign consistency.Compared with Lasso,SIS-Lasso,adaptive Lasso,it is found that the proposed method is better than other methods in terms of recovering the coefficient sign of linear part through the numerical simulation and real data analysis.