| Hastie and Tibshirani (1993) proposed the varying coefficient model, which is defined asY=αT(U)X+ε,where Y is the response variable, U,X=(X1,…,Xp)T are the associated covariates,α(·)=(α1(·),…,αp(·))T is a vector of unknown coefficient functions,αk(·)(k=1,…,p) are unknown coefficient functions, s is independent of (U,X)and satisfied with Eε=0,Var(ε)=σ2. The varying coefficient model is a new developmental direction in dealing with high dimension data of mathematical statistics in modern times, and it is widely used in many fields such as circumstance, biology and medicine etc.However, in practice, functional coefficients are rarely difficult to react on comprehensive and specific objective facts, which require parts of functional coefficients as constant coefficients. Namely, hypothesis H0:αi(·)=βi. is reasonable for somei. Then, under the condition of accepting H0, Fan and Zhang proposed the semi-varying coefficient modelY=αT(U)Z+βTX+ε,where Y is the response variable, U,X=(X1,…,Xq)T,Z = (Z1,…,Zp)T are the associated covariates,α(·)=(α1(·),…,αp(·))T is a vector of unknown coefficient functions,αj(·)(j=1,2,…,p) are unknown measurable functions from R to R ,β=(β1,…,βq)T is a vector by unknown ordinary coefficient,βk(k=1,…,q) are unknown parameters,εis independent of (U,X,Z) and satisfied with Eε=0,Var(ε)=σ2.Obviously ,ifαj(·) =0(j=1,…,p), the model becomes a linear model; if we regardβk(k=1,…,q)as some functions, the model means a varying coefficient model; if order j=1,Z1=1,then it becomes a partially linear model.We usually encounter such case, which is that there are some definite constraint conditions among variables, and this kind of model is called constraint regression model. Through the residual analysis, it indicates that errors have some daedal connection with each other, and constraint condition is likely to change correspondingly at random. So the hypothetical model of "equal variances and irrelevance" is of limitation and irrationality, therefore, on the basis of Lv's thought ,the condition of error correlation and random constraints are added on the semi-varying coefficient model. The new model that is called the mixed random constrained regression model of semi-varying coefficient model with co-relational error is as follows:where Y=(Y1,…,Yn)T is the response variable,α(·)=(α1(·),…,αp(·))T is a p dimension function vector,U,Z=(Z1,…,Zn)T,X=(X1,…,Xn)T are the associated covariates, and Xi=(xi1,…,Xip)T is component observed value ,β=(β1,…,βq)T is a vector of ordinary coefficient, A is a known matrix, b is a known vector,Σ1 and Z2 are known positive matrix,ε=(ε1,…,εn)T and e=(e1,…,en)T are random errors and independent with each other.Owing to "dimension curses", when U is high dimension data, parameter estimations appear terrible instability and the practical feasibility of models is bad. Thus, we generally suppose U to be an associated covariate of one dimension.In the paper there are three parts as follows:The first part is introduction. The development and research actuality of semi-varying coefficient models are mainly summarized. As well, the classical estimate methods are briefly reviewed and the semi-varying coefficient models in this paper are recommended.The second part discusses the robust M-estimation on semi-varying coefficient models.The robust M-estimate of the unknown measurable function is proposed by local linear method and weak congruence and the asymptotic normality are discussed, under the condition of hypothetical data with independent identical distribution and parameter vector without added constraint conditions. Based on the estimate of the measurable function, the generic M-estimates of the unknown parameter vector are given by Back-fitting technique and the asymptotic normality is discussed. In this process, the M-estimate method in the paper not only inherits the excellence of local linear method and LS method, but also obtains good robust.The third part discusses the random constrained estimation of semi-varying coefficient models with co-relational error. Firstly, the estimate of the unknown measurable function is proposed by local linear method. Secondly, the estimate of the unknown parameter is given by least-square technique. And the asymptotic normality of the estimation is investigated. The conclusion indicates that such parameter estimates of run-of-mill models have good asymptotic normality.
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