Perturbation Selection And Local Influence Analysis Of Density Estimation Based On Gaussian Kernel Function

The issue of nonparametric estimation for density function has been focused on for a long time and many methodologies have been proposed in this area. Among all the theories about nonparametric density function estimation, the kernel density estimator and the method of local polynomial log-likelihood have received a lot of attention, where the kernel density estimator is the foundation of many density estimators and the method of local polynomial log-likelihood can reduce the boundary bias. It is necessary to assess the influence of some inputs, such as the data points, on the density estimators though nonparametric methods are generally less sensitive to outliers than the parametric ones as well known.In this thesis, an approach of local influence analysis is proposed for the kernel density estimator and the method of local polynomial log-likelihood with Gaussian kernel functions used. This approach can be used to assess the influence of not only the data points but also some groups of data points close to each other, considering that both of the above two density estimators are based on the local smoothing. The proposed methodology is built on the basis of joint perturbation scheme for data points(or data groups) to avoid masking effect among the influential data points(or groups). None of the above two density estimators depends on the parametric likelihood and their inference results are both functions instead of vectors. Hence, none of the methods of local influence analysis based on the parametric likelihood(e.g. the ones using likelihood displacement function) or the methods for inference of vector type(e.g. generalize Cook’s statistics) can be directly used for these two density estimators. The proposed approach is based on a so-called density displacement function, which is a function of perturbation vector to measure the discrepancy between the estimated density functions with and without perturbation and can be viewed as the counterpart of the likelihood displacement in the scenario of density estimation. The concepts defined under likelihood displacement function, such as influence graph, perturbation direction, lifted line, normal curvature, influential direction, aggregate vector, can all be extended to the density displacement. In the proposed framework under the density function displacement, the influential direction or aggregate vector are still used as influence assessment statistics. For both the kernel density estimator and the method of local linear log-likelihood(as an example of local polynomial log-likelihood), the specific expressions for the normal curvatures of the lifted lines are derived and they turn out to be quadratic forms of the perturbation directions, which means the influential directions and the aggregate vectors can be easily obtained from the curvature expressions. Before the above procedure of local influence assessment, a strategy of perturbation selection based on metric tensor matrix of perturbation vector is used for assessing the appropriateness of the amount of perturbation introduced and adjusting the perturbation amount if necessary. It turns out that, under the scheme of joint weighting perturbation of data points, the original perturbation vector is appropriate, but under the scheme of joint weighting perturbation of data groups, the original perturbation vector is in inappropriate when the sizes of the groups are unbalanced and a transformation of the perturbation vector is used to adjust the perturbation amount.A simulation analysis is conducted to illustrate the proposed methodologies. In the simulated example, it seems that a single outlier produces slight influences on the density estimators, and its influence becomes much stronger when it is in a group of outliers close to each other. The influence assessment of the data groups also reveals such information.