Suffiffifficient Dimension Reduction Methods With Multivariate Response Variables Based On Cumulative Covariance Matrix

Measuring and testing the interdependence between two random variables has always been a fundamental problem in statistical research.In this thesis,a new robustness measure（Cumulative Covariance Matrix（CCM））,is proposed to test whether the conditional mean independence of a dimensional random vector y?R^q depends on another dimensional random vector x?R^p.We first prove that,tr{CCM（y|x）}30 holds,and that the equality sign holds when and only when the E（y|x）does not varies with x.We then investigate the asymptotic nature of the sample estimate and prove that if E（y|x）does not varies with X,then the corresponding sample estimator of Cumulative Covariance Matrix is consistent;otherwise,it is root-n consistent.We illustrate through extensive numerical Simulation Studies that the conditional mean independence test based on tr{CCM（y|x）}has good size and efficacy power,and this advantage is more significant in the presence of extreme values in the data.Finally,we propose an suffiffifficient dimension reduction method for multivariate responses data based on the Cumulative Covariance Matrix,The method is used to estimate the central dimensionality reduction subspace and the compatibility of the method is demonstrated.Our suffiffifficient dimension reduction method does not consider slicing,can avoid the instability of dimensionality reduction results due to multiple random projections,and has robust properties in the presence of outliers in response variables.We also demonstrate the finite-sample nature of our dimensionality reduction method through extensive numerical simulations and analysis of actual data on human blood pressure.The main structure of this thesis is as follows:Part Ⅰ:We propose the Cumulative Covariance Matrix to measure whether the conditional mean independence of a dimensional random vectory?R^q depends on another dimensional random vector x?R^p.This measure is robust in the presence of outliers or extreme values of the observed values of the covariates.We prove the asymptotic nature of its sample estimator.Part Ⅱ:We propose an suffiffifficient dimension reduction method for multivariate responses data on the cumulative covariance matrix suffiffifficient dimension reductionMethod for multivariate.The method is used to estimate the Central（Mean）Subspace S_y|xand prove the consistent of the method.Our suffiffifficient dimension reduction method does not consider slicing,can avoid the instability of the dimensionality reduction results due to multiple random projections,and has the property of being robust in the presence of outliers in the response variables.Part Ⅲ:Focusing on practical problems,we investigate experiments in which the response variable is not a scalar but a multivariate response variable.The analysis of actual human blood pressure data demonstrates the finite sample nature of our dimensionality reduction method.The research in this thesis is not only limited to theoretical studies,but also includes a large number of statistical simulations and actual data analysis,so that practice and theory can co-exist.