Variable Selection In Functional-coefficient Partial Modeling

In this paper, we consider the functional-coefficient partially linear model where Yi∈R is a real valued response variable, xi=(x1,...,xp)T, Ui=(Ui1,...,Uip)T,V=(V1,V2,…,Vq)T∈RT are the covariate variable,εi is a random error with E(εi,|xi,Ui,Vi)=0,D(εi)=σ2,θ(U)=(θ1,(U),…,θq(U))T, ε1,Xi, U1, V1are independent of each other; and varying coefficient functions are both unknown measurable functions, where is the sum of component function whose number is finite.In this paper, we mainly discuss the variable selection problem about the above mentioned model. First, we use the penalized least squares to achieve a objective function, and for the objective function, we take the penalized function like SCAD(Smoothly Clipped Absolute Deviation)(Fan and Li). Second, as and θ(Ui) are both of nonparametric, so we approximate each of them by the combination of basis functions. Then we obtain the estimators by minimizing the objective function. And with appropriate selection of the tuning parameters we show that this variable selection procedure can indentify the true model consistently, and the regularized estimators have oracle property.Structure of this paper:Chapter1, mainly show the context of the problem studied by the paper. And discuss the ideas and methods involved in solving the problem, also the conclusions acquired.Chapter2. Consider variable selection of nonparametric additive model as a special case of the varying coefficient partially linear model. Basis functions are used to approximate unknown functions in the model in the procedure of variable selection. Combined with the penalized least squares, we achieve a method of variable selection. In the method, we can make variable selection when estimates those nonparametric components.Chapter3. We make variable selection for the varying coefficient partially linear model by the method like dealing with variable selection of nonparametric components. In the procedure, those unknown nonparametric functions and unknown varying coefficient functions are approximate by basis functions. With the method, we can make variable selection when estimates those nonparametric components. This is an improvement of the two step variable selection method. Consider good properties of the regularized estimators depends on the choice of the tuning parameters, we improve the existing method of tuning parameters selection from the point that the model has the property of consistent.