Sliced Inverse Regression For Normal Mixture Distribution

Modern computer technology is enhancing our ability to deal with large-scale data.In a nonparametric regression,how to reduce the dimension of a high dimensional input variable with slight loss of information and obtain useful information is an important issue.Sufficient dimension reduction is an effective tool for this issue.The basic idea is to find the linear combination of the original independent variables without any assumption of parametric model.Among all the methods for sufficient dimension reduction,the sliced inverse regression(SIR)is basic and important.It can be easily implemented and is robust in most many scenarios.However,the sliced inverse regression requires the linear design condition,which almost means that independent variables follow elliptically symmetric distribution.Unfortunately,in practice,this condition is sometimes violated.In this thesis,we propose a new approach to improve the sliced inverse regression so that it can be applied when the indicator vector follows a multivariate normal mixture distribution.Under these situations,we design a kernel matrix based the idea of sliced inverse regression and two-step expectation and prove that column space of this kernel matrix is a subspace of the dimension reduction space.In addition,by treating the component markers of the normal mixture distribution,we give the estimations of the kernel matrix,the dimension reduction directions and structural dimension.Furthermore,we give an algorithm for the above procedure.Simulation studies are conducted to examine the performance of the proposed methodologies.In these simulation studies,we find that our method is suitable for not only the cases with odd regression function but also the ones with even function,and it can enhance the accuracy of the estimations.