The Regression Analysis Of Varying-Coefficient Models With Censored Data

the nonparametric regression analysis is one of the leading methods of the modernstatistical analysis,It needs few hypotheses and its main advantage is to be very robust. So it is widely applied.The essence of the the nonparametric regression analysis methods is local estimator or local smoothing technique.In general,the nonparametricregression function is well estimated by the nonparametric regression analysis methods when the variable X is one dimension.But the multivariable nonparametricregression could not be well estimated by the local estimators because there is only a little data in the local fields of the high dimension regression variable X.This phenomenon is said to be 'the curse of dimension' .Due to a lot of the high dimension data is often happened,the analysis of high dimension data is one of the aspects in which a lot of statisticians are interested.However,varying-coefficient models is a effective method to solve the problem of 'the curse of dimension' .In this paper,we consider varying-coefficient models in this formwhere a(·) = (a₁(·),…,a_p(·))^T, a₁(u),…,a_p(u) is unspecified smoothing function that needs to estimate,Y is real variable , X = (X₁,…, X_p)^T is random vetor, U is random variable,and its density function is f(u), e is random error, and E(ε|U, X) = 0,V ar(ε|U,X)=σ²(U,X).However,in the real problems,for example ,in the fields of reliable lifespan experiment,medicine track,survival analysis and so on,Y can not be observed because it is censored.let C denotes the censoring random variable, Y and C are independent random variable under the condition that U and X are given. T = min(Y, C), T_i= min(y_i,c_i),δ_i =I(y_i≤c_i)(i = 1,2,…, n), where I(·) denotes the sign function of a event, if Y is not censored ,thenδ= 1, if Y is censored, then (δ= 0。We can only observe {([U_i, X_i, T_i,δ_i); i = 1,…, n}。Because responding variable are censored,we can not use the methods directly which we use on full data.So we should transform the censored data.In this paper,Class-K method is used to transform data.we transform data (U, X, T,δ) to data (U, X, Y^*),whereφ₁(…) andφ₂(…) are transformation functions, and E(Y*|U, X) = E(Y|U, X).In this paper,the estimation of model(*) with censored data is constructed by local polynomial regression and Class-K method.And we respectively discuss model(*) whose different coefficient functions admit the same degree of smoothness and different degrees of smoothness in detail.First,if different coefficient functions of model(*) have the same degree of smoothness,we estimate model(*) by local linear function. Asymptotic bias and variance are obtained,and asymptotic normality is also established.Second, different coefficient functions of model(*) admit different degrees of smoothness.if we use ordinary estimation methods to estimate it,we can not achieve the optimal rate of convergence. Then we estimate coefficient functions with higher degree of smoothness by local m degree polynomial,and we estimate others by local linear function.The optimal rate of convergence can be achieved and asympt tic bias and variance are all obtained.