Variable Selection And Applications For Linear Models And Partially Linear Models

Linear model is a simple and common kind of parametric models.Due to the merits of concise form and good interpretability,linear model has gained tremendous popularity among the researchers.Meanwhile,partially linear model is a distinctly important kind of semiparametric models.Based on linear model,partially linear model further adds the nonparametric part.Thus,partially linear model not only inherits the good interpretability of linear model and retains the flexibility of nonparametric models,but also conquers the problem of “dimensionality curse” that encountered by nonparametric models.This paper primarily studies the problems of variable selection and applications in linear models and partially linear models.To be specific,the research content consists of the following four parts.Chapter 2 studies the problem of simultaneous nonnegative estimation and variable selection in sparse high-dimensional linear models.Based on adaptive elastic-net penalty function,we propose the nonnegative adaptive elastic-net estimator under the nonnegative constraint.Under suitable regularity conditions,we establish the variable selection consistency and asymptotic normality of the proposed estimator.In addition,we provide the specific algorithm for the proposed estimator.Finally,numerical simulations and a real data analysis are carried out to verify the finite sample efficiency of the proposed method.In Chapter 3,we further study the problem of simultaneous nonnegative estimation and variable selection in sparse high-dimensional linear models.To improve the biasness of the nonnegative convex penalized methods,we propose the nonnegative MCP estimator with the aid of MCP penalty function under the nonnegative constraint.We prove that the proposed estimator enjoys the oracle property with less regularity conditions than other methods.Moreover,we develop a composite algorithm,which is a coupling of the difference convex algorithm with the multiplicative updates algorithm,for the implementation of the objective function of the proposed estimator.Finally,the results of numerical simulations and a real data analysis confirm the superiority of the proposed method over other methods.Chapter 4 studies the robust estimation and variable selection in partially linear models with fixed number of covariates.In order to achieve robustness against outliers both in the response and covariate domain,we combine weighted LAD regression with adaptive Lasso penalty to achieve simultaneous robust estimation and variable selection of the linear parameters.In addition,we estimate the unknown smooth function of the nonparametric part by a robust local linear regression.Finally,numerical simulations and a real data analysis are carried out to verify the finite sample performance of the proposed method.In Chapter 5,we study the robust estimation and variable selection in partially linear models with a diverging number of covariates.We devote to extending the local modal regression to partially linear models so as to obtain simultaneous robustness and efficiency for different error distributions.We show that the proposed method is more efficient in the presence of outliers or heavy-tail error distributions,and as asymptotically efficient as the corresponding least squares regression when there are no outliers and the error distribution is normal.Meanwhile,we also consider the problem of variable selection for the linear part with the aid of SCAD penalty fuction.Furthermore,we discuss the optimal bandwidth in theory and the selection of bandwidths in practice.In addition,when the number of covariates is larger than the sample size,we propose a two-step variable selection procedure to deal with high-dimensional data.Finally,numerical simulations and a real data analysis are carried out to evaluate the robustness and efficiency of the proposed method.