Wavelet. Semiparametric Regression Model Are Estimated

This paper studies the theory and application of semiparametric regressionmodel under fixed design: yi = xiβ +g(ti)+ εi ,i = 1,2,…n, Where xi = h(ti) + vi,Eεi =0, i = 1,2, …n. By using the least square method and the weighted functionmethod of nonparametric estimation together, this paper defines the wavelet estimators of β and g(·). But the weighted function here is not in the common weighted functions, which include kernal weighted function and nearest neighbor weighted function. However, it is a kind of weighted function deduced by wavelet method, so we call it wavelet weighted function here. On the basis of the above, as to several rare situations, this paper established some new research and discussion and obtained satisfactory results, which is divided into three Specific parts as following.The first part: since the semiparametric regression model is usually studied when the error sequence has the same variance, this chapter deals with a special heteroscedasticity situation, namely, εi= σiei , where σi2=f(ui), {ei}is an i.i.d. random error sequence with mean 0 and variance 1. Under the above consumption, the wavelet estimators of β and g(·). are defined. Moreover, their weak consistency, strong consistency and asymptotic normality are proved.The second part: when {εi} is an i.i.d. random error sequence with mean 0 and variance σ2, on the basis of the results about the wavelet estimators ofβ ,g(·) and a , we define the Bootstrap statistics for β n of βn and for σn*2 of σ2, where βn and σn2 are the wavelet estimators of β and σ2. Then it is proved that under the given original sample, the asymptotic distributions of n(β*n-βn) and n(σ*n 2-σ2n) are same as the ones of n(βn - β) andThe third part: When { εi} is an independent random error sequence with mean 0, the wavelet estimators of β and g(·) are constructed under censored data. And their consistency and asymptotic normality are obtained.