Local M-Estimation For Varying-Coefficient Partially Linear Model

In nonparametric regression, the methods include kernel-type estimation, local polynomial estimation, smoothing spline estimation, series estimation and so on. These methods deal well with one-dimensional date, however, with the increase of dimension, the samples we can get become very sparse and the estimation will be instable, namely, "curse of dimensionality", which results in that the conventional methods in nonparametric regression show uselessness to high- dimensional date. Many powerful approaches have been created to avoid the so-called "curse of dimensionality", including varying-coefficient model. Varying-coefficient model not only inherits the traits of robust from nonparametric models, but also keeps the advantages of linear models. Therefore, the researchers have been paid more attention about it recently and varying-coefficient model has been widely used in biomedical, epidemiological, environmental science and other fields.Varying-coefficient model was introduced by Hastie and Tibshirani in 1993. It is an abstract model, so the practicality of it is poor. Many researchers deal with it according to circumstances. For example, Zhang and Wang (2005) proposed varying-coefficient partially linear model which is derived from varying-coefficient model. It is a kind of varying-coefficient model that the constant part function and the coefficient functions have different variables and it is widely applied in practice. Zhang and Wang (2005) adopted the local polynomial technique and got estimations of the constant part function and the coefficient functions of varying-coefficient partially linear model. The weak consistency and asymptotic normality of estimation on the constant part function and the coefficient functions are given under the condition which the sample is independent identically distributed.In this paper, we apply the local M-estimation augmented with variable bandwidth and get estimations of the constant part function and the coefficient functions. The weak consistency and asymptotic normality of estimation are given and proofed under the condition which the sample is independent identically distributed. The local M-estimation inherits all advantages of local polynomial regression method. Moreover, the local M-estimation can achieve desirable robustness properties. The local M-estimation with variable bandwidth is similar to the local M-estimation methods and is enhanced via incorporating a variable bandwidth scheme. This allows the resulting estimation procedure to cope well with spatially inhomogeneous curves. To test the effect of estimation, we give a specific example and simulate it, The simulated results showed that those methods are fairly ideal.