Two-Stage Regression Estimation For Semiparametric Estimation Regression Models

Semiparametric regression model is a kind of important statistical model which develops from the 1980's . This kind of model contains both parametric and nonparametric components and can be used to describe many practical problem. So it draws attentions widely. The form of a semiparametric regression model is deined as:where X_i are p-dimensional design variables . is a vector of unknown parameters. g(·) is an unknown Borel function in R¹. X_i^Tβ , g(t_i) are called the parametric component and the nonparametric component of this model, respectively.In the related literatures, most work on the semiparametric regression follow this way, one of the most typical example is the kernel estimators. In the first step, β is given, We use a nonparametric regression model based on to estimate g, called gλ(·,β); The next step we use gλ to replace g and the following least squares estimators method to find the solution β.However, the final estimate of g is gλ(·,β), Here A is the smooth parameter in nonpa-rameter regression estimation. For example, A is the bandwidth h for kernel estimators. Methods commonly used in the calculation of bandwidth include CV methods, Risk Estimation and GCV methods. In lack of a uniform bandwidth selection criterion, the selected bandwidth are optimal for different criterions which need intensive calculation. Sothe performance of nonparametric component g(·)'s estimation has direct impact on the parameter estimates of β. Therefore, the estimation of the nonparametric component g(·) is particularly important. But thestandard nonparametric regression estimators have no a satisfactory convergence rate especially for multivariate models. To improve the non-parametric estimation in the sense of convergence rate, this paper proposes a two-stage regression estimation, which combines nonparametric regression with parametric regression.In the first-stage, some nonparametric regression estimators with different band-widths are prepared as prior selections. These selected estimators are in the second-stage combined by a parametric regression technique so as to forma new estimator. The optimal design conditions, including optimal bandwidths, are obtained. Thus we avoid the use of approaches like CV methods to choose a bandwidth, and it reduces the amount of calculation. For univariate models, the optimal convergence rate of the mean squared error is of order O(n^-8/9) at the interior points which is not achieved by the existing methods.Our new method is suitable for general nonparametric regression models regardless of the climension of explanatory variable and the structure assumption on regression function. Some simulations are given to evaluate the finite-sample performance of the new method and compare it with the existing methods.when the above semiparametric regression model satisfies some conditions,we drive the main result of the paper:Conclusion 1:The bias of the new estimator g(t) is of order O(n^-4/9). however, the original one is of order O(n^-2/5), which is smaller than that of the original one. And we can see that variance of the new and original estimators have the same asymptotic order, The mean square convergence rate of our new method reaches the order of O(n^-8/9), while the original is of order O(n^-4/5). We can see that the new estimator improves the original one very much in the sense of convergence rate.Conclusion 2:Our method obtained the optimal bandwidth through a two-stage estimation. Thus we avoid the use of approaches like CV methods to choose a bandwidth and it reduces the amount of calculation.