Two-Stage Regression Estimate For Local Polynomial Model

Local polynomial regression technique is a popular nonparametric regression technique. The method adapts to various types of designs such as random and fixed designs, highly clustered and nearly uniform designs. Furthermore, there is an absence of boundary effects: the bias at the boundary stays automatically of the same order as in the interior, without use of specific boundary kernels, which is remarkably different from the other methods. It was systematically studied by Fan(1992,1993a)[1,2],Fan and Gijbels (1992)[3]and Rubbert and wand(1994)[4].However, local polynomial regression resolves lots of questions, it has not a satisfactory convergence rate. And at the same time, it is a hard work to find the optimal bandwidth. The different criteria lead to different bandwidths, and the different bandwidths lead to different results. Now, there are a lot of methods for getting bandwidth such as CV, GCV and so on. As we all know, Cross-validation is a popular data-driven method , but it need huge calculation. In this paper, our technique allows us to select appropriate bandwidth more easily while keeping the effectiveness of local polynomial regression. To improve local polynomial regression in the sense of convergence rate, we propose a two-stage regression estimation. In the first stage, we get estimation without thinking of bandwidth. The result change with bandwidth. These selected estimators are in the second stage combined by a parametric regression techniques so as to form a new estimator. As a result, for univariate models, the optimal convergence rate of the mean squared error is of order the O(n^{-(2p+6)/(2p+7)}),O(n^{-(2p+8)/(2p+9)}), which is of order O(n^{-((2p+2)/(2p+3))},O(n^{-((2p+4)/(2p+3))} achieved by the existing methods. As we can see, approximated mean squared error is derived after using new method.In this paper,our technique allows us to select appropriate bandwidth easier while keeping the effectiveness of local polynomial regression. At last, some simulations are given to illustrate the theoretical results and compare it with the existing methods.