| Varying coefficient models has important position in modern Statistics.It is promotion of classical linear regression model, proposed by Hastie andTibshirani [1] and defined by the following linear model: Y=sum from j=1 to p+mαj(U)Xjε,for given covariates X=(X1,X2,...,Xp+m)T and U=(U1, U2,…, Uq)T,Y is response variable,αj(U)(j=1,2,…,p+m) is unknownfunction,εis random error, E(ε|U, X1,X2,...,Xp+m)=0 andVar(ε|U, X1,X2,..., Xp+m)=σ2(U). Moreover, it is well-recognized thatthe model has extremely wide applications. For example, see Hoover et al[2] for novel applications of the model to longitudinal data; Fan, Yao andCai [3] for application in ecologic data analysis; Chen and Tsay [4], andCai,Fan and Yao [5] for nonlinear time series applications.In practise, we often want to know if a covariate affects the response orif the coefficients are really varying, see Fan and Zhang [6]. This amounts to test if the whole coefficient is zero or constant, namely, testing the nullhypothesis H0 thatαj(U)=αj, for some functions. The model under H0will be called a semi-varying coefficient model: Y=sum from j=1 to pαj(U)Xj+sum from j=p+1 to p+mαjXj+εfor given covariates X=(X1,…, Xp, Xp+1,..., Xp+m)T, U=(U1, U2,…,Up)T and Y is response variable,αj(U)(j=1,2,...,p) is unknownfunction,αj(j=p+1,...,p+m) is constant,εis random error, f(u)be the marginal density of U, E(ε|U, X1,...,Xp,...=0 andVar(ε|U, X1,...,Xp,...,Xp+m)=σ2(U). This model consists of a non-parametric part that involves coefficient functionsαj(U)(j=1,2,...,p)and a linear part that involves constant coefficientαj (j=p+1,…, p+m).If the constant coefficientαj(j=p+1,..., p+m) is viewed as a function,the semi-varying coefficient model can be regarded as a special case of thevarying coefficient model.Wenyang Zhang, Sik-Yum Lee and Xinyuan Song [7] have discussedthe estimation and simulation under the some degree of smoothness ofcoefficient functions in the semi-varying coefficient model. Following, un-der the condition that the coefficient functions possess different degrees ofsmoothness, a two-step method is proposed. In addition, we show thatone-step method can not be optimal when different coefficient functionsadmit different degrees of smoothness. This drawback can be repaired byusing our proposed two-step estimation procedure.
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