Local Impact Analysis Of PHD Method In Full Dimension Reduction Theory

In the development of statistics, we are faced with more and more challenges of high dimensional data. Using nonparametric methods to deal with high dimensional data may suffer from “curse of dimension”. Especially in the regression, when the basic form of the model is unknown and a lot of predictor candidates exist,nonparametric methods such as kernel function and spline approximation may all be inefficient due to the sparse data. However, if the response only depends on several linear combinations of the predictor candidates, we only need to find out the coefficients of these linear combinations to solve the problem of high dimension.However, it is a difficult problem to estimate the combination coefficients without any assumption about the relationship between the response and predictors. The theory of sufficient dimension reduction is to solve this problem. This issue has drawn a lot of attention and a variety of methods and theories have been developed for it.The approach of principal Hessian direction is one of the most extensively applied methodologies of sufficient dimension reduction. On the other hand, it is also a method which needs a strict condition about the distribution of data, since its theoretical basis is Stein lemma which depends on the normal assumption for predictor vector. Hence, influence analysis of this method is necessary. In this fiend,there is now only a case-deletion approach based on the influence function. The efficiency of this method for detection of special influence structures is unsatisfactory.Therefore, a methodology of local influence analysis for principal Hessian directions is proposed in this article. This theory is based on a so-called space displacement function and quasi-curvature and can be used under joint perturbation scheme. It is more powerful than the case-deletion method, as to detection of influential aspects in the model.The following are theoretical results obtained in this article. For both y-PHD which is directly based on the response and r-PHD that is related to the least square residual, the specific expressions of the quasi-curvature and influential direction are obtained under the scheme of perturbation to data points, so the statistic of local influence assessment is constructed. Moreover, it is found that the proposed methodology is invariant under the affine transformation of the predictor vector.Considering the background and purpose of sufficient dimension reduction, this invariance property is quite necessary and important. In addition, we apply the proposed method for a simulation study and a real data analysis, illustrating its advantage by the comparison with case-deletion method.