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    Sufficient Dimension Reduction with Missing Data

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    Genre
    Thesis/Dissertation
    Date
    2017
    Author
    XIA, QI
    Advisor
    Dong, Yuexiao
    Tang, Cheng-Yong
    Committee member
    Wei, William W. S.
    Han, Xu
    Yang, Yang
    Department
    Statistics
    Subject
    Statistics
    Missing at Random
    Ordinary Least Squares
    Principal Hessian Directions
    Sliced Inverse Regression
    Sufficient Dimension Reduction
    Permanent link to this record
    http://hdl.handle.net/20.500.12613/3859
    
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    DOI
    http://dx.doi.org/10.34944/dspace/3841
    Abstract
    Existing sufficient dimension reduction (SDR) methods typically consider cases with no missing data. The dissertation aims to propose methods to facilitate the SDR methods when the response can be missing. The first part of the dissertation focuses on the seminal sliced inverse regression (SIR) approach proposed by Li (1991). We show that missing responses generally affect the validity of the inverse regressions under the mechanism of missing at random. We then propose a simple and effective adjustment with inverse probability weighting that guarantees the validity of SIR. Furthermore, a marginal coordinate test is introduced for this adjusted estimator. The proposed method share the simplicity of SIR and requires the linear conditional mean assumption. The second part of the dissertation proposes two new estimating equation procedures: the complete case estimating equation approach and the inverse probability weighted estimating equation approach. The two approaches are applied to a family of dimension reduction methods, which includes ordinary least squares, principal Hessian directions, and SIR. By solving the estimating equations, the two approaches are able to avoid the common assumptions in the SDR literature, the linear conditional mean assumption, and the constant conditional variance assumption. For all the aforementioned methods, the asymptotic properties are established, and their superb finite sample performances are demonstrated through extensive numerical studies as well as a real data analysis. In addition, existing estimators of the central mean space have uneven performances across different types of link functions. To address this limitation, a new hybrid SDR estimator is proposed that successfully recovers the central mean space for a wide range of link functions. Based on the new hybrid estimator, we further study the order determination procedure and the marginal coordinate test. The superior performance of the hybrid estimator over existing methods is demonstrated in simulation studies. Note that the proposed procedures dealing with the missing response at random can be simply adapted to this hybrid method.
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