Local Influence Analysis Of Average Third-Order Moment Estimation In Dimensionality Space Slices

In multivariate non-parametric regression,it may happen that the response is related to the predictors through only a small number of their linear combinations.In that case,finding out those linear combinations of the predictors can reduce the dimension of the regression and may bring some improvements of the regression such as the enhancement of fitting accuracy,visualization,and so on.A variety of sufficient dimension reduction approaches were proposed to handle this issue.Among them,the sliced inverse regression and sliced average variance estimation method are frequently used.However,in the scenarios where both the inverse conditional mean and variance are constant,none of these two methods can be used.To deal with this problem,a method,called sliced average third-moment estimation,was proposed and received a lot of attention.This method depends on the estimation of the conditional third-moment of the predictor vector,so it is necessary to consider the sensitivity analysis of this method.In this thesis,we focus on the local influence analysis for central subspace estimator given by the sliced average third-moment estimation method.The proposed methodology of local influence analysis for the sliced average thirdmoment estimation is based on a so-called space displacement function,which is used to measure the difference between the central space estimators with and without perturbation introduced into the model.A framework of the local influence analysis for the sliced average third-moment estimation method is constructed,where the expressions of all the key quantities such as so-called quasi-curvature and influential direction can be obtained.Under this framework,the local influence assessment statistic,influential direction,can be easily obtained by maximizing the quasi-curvature which can be expressed as a quadratic form of the perturbation direction.Hence,the computational burden for this method is slight.Furthermore,to assess the impact of the observations on the central subspace estimator,we design a perturbation scheme,under which the specific expressions of the quasi-curvature and influential direction are derived.To illustrate the proposed method of local influence analysis,we apply it to the data simulated from a classical model where the inverse conditional mean and variance of the predictor vector are constant.Under this model,the sliced average third-moment estimation method performs well,but none of the sliced inverse regression and sliced average variance estimation method works.The simulation results show that the proposed method of local influence analysis can successfully identify the artificial outliers,and some new interesting insights are found.