Perturbation Selection And Local Impact Analysis Of Nonparametric Regression Based On Gaussian Kernel Function

The issue of nonparametric regression has been focused on for a long time and many methodologies have been proposed in this area. Among all the theories about nonparametric regression, Nadaraya-Watson kernel estimator and the local polynomial regression have received a lot of attention, where the Nadaraya-Watson kernel estimator is the foundation of many kernel regression methods and the local polynomial regression can reduce the boundary bias of the regression function estimates. It is necessary to assess the influence of some inputs, such as the data points, on the estimators of regression function though nonparametric regressions are generally less sensitive to outliers than the parametric ones as well known.In this thesis, an approach is proposed for local influence analysis of the Nadaraya-Watson kernel estimator and the local polynomial regression with Gaussian kernel functions. As the local smoothing technique play key roles in both of the above two nonparametric regression methods, the proposed approach is designed for assessing the influence of not only the data points but also some groups of data points with regressor values close to each other. To avoid masking effect among the influential data points or groups, the proposed methodology is constructed under joint perturbation scheme for data points or data groups. None of the above two nonparametric regression methods depend on any likelihood function and their inference results are not of vector type but functions. Hence, none of the methods employing likelihood displacement or the methods for inference of vector type, say the ones using generalize influence function, is suitable for the local influence analysis of these two regression methods. The proposed approach is based on a so-called regression displacement function, which is a func tion of perturbation vector to measure the discrepancy between the estimated regression functions with and without perturbation and can be viewed as the counterpart of the likelihood displacement in the scenario of nonparametric regression. The concepts de fined under likelihood displacement function, such as influence graph, perturbation direction, lifted line, normal curvature, influential direction, aggregate vector, can all be extended to the regression displacement function. In the framework based on the regression displacement, the influential direction or aggregate vector are still used as influence measure statistics. For both Nadaraya-Watson kernel estimator and the local linear regression(as an example of local polynomial regression), the specific expressions for the normal curvatures of the lifted lines are derived and shown to be quadratic forms of the perturbation directions. That means the influential directions and the aggregate vectors can be easily obtained from those expressions. A metric tensor matrix of perturbation vector is used for perturbation selection including the assessment for the appropriateness of the perturbation and the adjustment of the perturbation if necessary. Assuming that the data points are identically and independently distributed, the original perturbation vector is shown to be appropriate under the scheme of joint weighting perturbation of data points, but under the scheme of joint weighting perturbation of data groups, the original perturbation vector needs to be transformed if the sizes of the groups are unbalanced. A simulation study is conducted to illustrate the proposed methodologies.