Improvement Of SIR-?and SAVE Under Multivariate Normal Mixture Distribution

Regression or classification problems with high-dimensional predictors are increasingly popular in contemporary applications.An important problem that arises from these applications is how to avoid smoothing in a high-dimensional space,which seriously hinders the accuracy of statistical inference — a phenomenon commonly known as the curse of dimensionality(Bellman,1961).Sufficient dimension reduction estimates,especially those based on inverse conditional moments,can avoid the curse of dimensionality.Hence,sufficient dimension reduction is also 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 the 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 well-known solutions for finding effective dimension reduction(edr)directions under even regression function,sliced average variance estimator(SAVE)and the second type sliced inverse regression(SIR-II)are two effective and very easily implemented methods in most cases of sufficient dimension reduction.In this thesis,we investigate whether and how SAVE and SIR-II are applicable with multivariate normal mixture distributions and if not,how to improve the two methods.Simulation indicate that the above two methods are not suitable for the scenarios where the predictor vector follows normal mixture distributions.We propose new methods to improve SAVE and SIR-II.Simulation studies are conducted to examine their performance when the predictor vector follows normal mixture distributions.