Variable Selection Of Two Semi-parametric Models Based On Exponential Squared Loss

Semi-parametric models not only have the advantages of easy interpretation of parametric mode.l,but also have the flexibility of non-paramet.ric model.When the dimension of the covariable is high,the semi-parametric model can also overcome the problem of "curse of dimensionality" of the non-parametric model.Therefore,this kind of model has attracted extensive attention of many scholars,and has been widely used in many fields such as economics and biology.In this paper,we mainly study the estimation and variable selection of two kinds of semi-parametric models:partially linear a.dditive model and varying coefficient partially nonlinear model.It is well known that most of the existing estimat,ion methods are based on least squares method.However,this method is very sensitive to the outlier or heavy-tailed error distribution in data,and is not robust,which greatly reduces the validity of the estimation.This requires us to find a more robust estimation method.In this paper,we shall use the exponential square loss method to estimate.For the problem of variable selection,on the one hand,we hope to select only a few covariates which are really related to the response variable in the model to achieve a good predict.ion effect.On the other hand,we hope that the variable selection method we used is more robust,and when there are outliers or heavy-tailed errors in data,t,he result of variable selection will not be greatly affected.Therefore,based on exponential squared loss,this paper uses SCAD penalty function to select variables for two semi-parametric models.This paper is divided into four chapters.In the first chapter,we introduce the basic knowledge of partially linear additive model,varying coefficient partially nonlinear model and exponential square loss method,as well as their correspond-ing research background and current situation.The second and third chapters are the main work of this paper.In Chapter 2,the robust estimation and variable selection of partially linear additive models are studied.The additive part of the model is approximated by B-spline basis function and the SCAD penalty function method is used to select the variables of the model.All this work is based on the exponential square loss method.Under proper regular conditions,we establish and prove the theoretical properties of the estimation and variable selection of the methods used.In addition,in numerical simulation of example 1,we use e the mean square error(MSE)to evaluate the validity for parameter part and average square error(RASE)to evaluate the validity for additive part.Compared with other methods,our method achieves good estimation results.In Example 2,RASE is used to evaluate the validity of the estimated function for the additive part,and the generalized mean square error(GMSE)is used to evaluate the validity for the parameter part.When the sample size is 200 and the error distribution is N(0,1),the penalized exponential squared loss(PESL)RASE is slightly larger than the penalized least squares estimator(PLSE)?that is,the effectiveness of PESL's non-parametric estimation is slightly worse,but correct fit(CF)is larger,that is,the effectiveness of variable selection is better.In the case of other error distribution at the sample size of 200,PESL's RASE and CF are always superior to the other three methods.When the error is subject to N(0,1)and t(3),the CFs of PESL with the sample size of 200 are less than those of the penalized modal regression(PSME),but the GMSEs of PESL are smaller than those of PSME,and PESL performs well under the other error distributions.With the increase of sample size,when the sample size is 400 and 600,the RASE,GMSE and variable selection of PESL are better than the other three methods.And when the error distribution obeys the contaminated normal distribution,with the increase of sample size,the superiority of PESL becomes more and more obvious,which shows that PESL is a robust and effective estimation method.In the real data analysis,we find that the exponential square loss method can select important variables and has better prediction effect than other methods.Chapter 3 uses exponential square loss method to select the variables for the varying coefficient partially nonlinear model-The varying coefficient part of the model is approximated by B-spline basis function,and the variable selection of the model is performed by SCAD penalty function.Under certain regular conditions,the theoretical properties of variable selection are established and proved.The fourth chapter is the summary of this paper and the prospect of future work.