Local Impact Analysis And Robustness Of SAVE And SIR - Ⅱ Estimators In The Theory Of Full Dimensional Reduction

Sufficient dimension reduction is an important issue in the field of nonparametric regression. It focuses on dimension reduction of regressors by finding out a small number of linear combination of the original regressors which can replace the original regressors in the regression without any loss of information. Among all the theories about sufficient dimension reduction, the methods of sliced average variance estimate(SAVE) and the second type sliced inverse regression(SIR-II) have received a lot of attention. Both of these two methods can avoid the problem that the centered inverse regression curve degenerates for some symmetric response curve, which happens to the method of sliced inverse regression(SIR), a classical approach for sufficient dimension reduction. As their inference results, the dimension reduction space estimates, will be used in the nonparametric regression, their robustness properties are crucial. Influence analysis is a useful method of statistical diagnostic. The existing influence analysis of these two dimension reduction methods is about influence function, where the influence of data points is assessed by case-deletion method which may suffers masking effect.In this thesis, an approach of local influence analysis is proposed for the sliced average variance estimate and the second type sliced inverse regression. The proposed methodology is built on the basis of a joint perturbation scheme for data points to avoid masking effect among the influential cases. None of the above two sufficient dimension reduction methods depends on the likelihood and their inference results are both functions instead of vectors. Hence, none of the approaches based on the likelihood displacement or those methods for inference of vector type(e.g. generalize Cook’s statistics) can be directly used for the local influence analysis of these two dimension reduction procedures. The proposed approach is based on a so-called space displacement function, a function of perturbation vector which measures the discrepancy between the dimension reduction space estimates with and without perturbation and can be viewed as the counterpart of the likelihood displacement in the scenario of sufficient dimension reduction. Some concepts in the framework of likelihood displacement, including influence graph, perturbation direction and lifted line are extended to the space displacement. In the framework of space displacement, the counterpart of normal curvature is called quasi-curvature and the perturbation direction maximizing it is defined as influential direction which is used as the influence assessment statistic. For both the sliced average variance estimate and the second type slice inverse regression, the specific expressions for the quasi-curvatures of the lifted lines are derived and are proved to be quadratic forms of the perturbation directions, which means the influential directions can be easily obtained from the quasi-curvature expressions. The proposed methods of local influence analysis are shown to be invariant under the invertible affine transformation of the predictor vector. A simulation analysis is conducted to illustrate the proposed methodologies.