| Given a large number of predictors in a regression, it is often desirable to reduce the dimensionality of the problem by replacing the original high-dimensional data with a low-dimensional space composed of a few key predictors or linear combinations of predictors. The theory of sufficient dimension reduction ( SDR) targets the reduction of dimension without losing any information on the conditional distribution of the response given the predictors, and without pre-specifying any parametric model. A large number of sufficient dimension reduction approaches have been developed that provide estimates of the key linear combinations of all predictors. However, none of the existing approaches address the issue of whether or not all of the predictors are needed. In addition, most existing SDR approaches require a critical assumption, the linearity condition, on the marginal distribution of predictors. In this thesis, I first propose a set of model-free variable selection methods within the framework of sufficient dimension reduction. The proposed approaches assume no model prior to selection, require no nonparametric smoothing, and predictor effects are not confined to the conditional mean. Given a set of key predictors, the second part of the thesis targets the identification of the key linear combinations of predictors, which preserve all characteristics of the regression. Specifically we propose two extensions to existing SDR estimation approaches, SIR3 and cluster-based ordinary least squares, that do not require the linearity condition. The thesis concludes with a discussion of future research work. |
