Stable and pseudo sufficient dimension reduction
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Due to recent advances of modern technology, abundant data are collected in many scientific areas. The developments in theory and methodology for sufficient dimension reduction (SDR) have provided a powerful tool to study such high dimensional data. Most existing methods are aiming at estimating the basis matrix and structural dimension of the central subspace (CS). This dissertation is composed with three parts. In the first study, we introduce stable estimation procedures for several aspects of a sufficient dimension reduction matrix, including a stable method for estimating structural dimension, a Grassmann Manifold sparse estimate for the CS, a stable nonsparse estimate for the CS. In the second study, in order to obtain a reliable estimate for correlated predictors, we uncover the underlying relationship between ridge regression and measurement error regression. With such a connection, we propose a general SDR estimation procedure to obtain an estimate from a different subspace instead of the targeted population parameter space. In the third study, we combine the stable and pseudo approach together and tackle the small n large p problem for dimension reduction. Theoretical results are established for our methods and the efficacy of the proposed methods is demonstrated by simulation studies and real data analyses.