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Sparse Minimum Discrepancy Approach to Sufficient Dimension Reduction with Simultaneous Variable Selection in Ultrahigh Dimension

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  • Wei Qian
  • Shanshan Ding
  • R. Dennis Cook

Abstract

Sufficient dimension reduction (SDR) is known to be a powerful tool for achieving data reduction and data visualization in regression and classification problems. In this work, we study ultrahigh-dimensional SDR problems and propose solutions under a unified minimum discrepancy approach with regularization. When p grows exponentially with n, consistency results in both central subspace estimation and variable selection are established simultaneously for important SDR methods, including sliced inverse regression (SIR), principal fitted component (PFC), and sliced average variance estimation (SAVE). Special sparse structures of large predictor or error covariance are also considered for potentially better performance. In addition, the proposed approach is equipped with a new algorithm to efficiently solve the regularized objective functions and a new data-driven procedure to determine structural dimension and tuning parameters, without the need to invert a large covariance matrix. Simulations and a real data analysis are offered to demonstrate the promise of our proposal in ultrahigh-dimensional settings. Supplementary materials for this article are available online.

Suggested Citation

  • Wei Qian & Shanshan Ding & R. Dennis Cook, 2019. "Sparse Minimum Discrepancy Approach to Sufficient Dimension Reduction with Simultaneous Variable Selection in Ultrahigh Dimension," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 114(527), pages 1277-1290, July.
  • Handle: RePEc:taf:jnlasa:v:114:y:2019:i:527:p:1277-1290
    DOI: 10.1080/01621459.2018.1497498
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    Cited by:

    1. Fang, Fang & Yu, Zhou, 2020. "Model averaging assisted sufficient dimension reduction," Computational Statistics & Data Analysis, Elsevier, vol. 152(C).
    2. Qin Wang & Yuan Xue, 2023. "A structured covariance ensemble for sufficient dimension reduction," Advances in Data Analysis and Classification, Springer;German Classification Society - Gesellschaft für Klassifikation (GfKl);Japanese Classification Society (JCS);Classification and Data Analysis Group of the Italian Statistical Society (CLADAG);International Federation of Classification Societies (IFCS), vol. 17(3), pages 777-800, September.
    3. Weng, Jiaying, 2022. "Fourier transform sparse inverse regression estimators for sufficient variable selection," Computational Statistics & Data Analysis, Elsevier, vol. 168(C).
    4. Wang, Qin & Xue, Yuan, 2021. "An ensemble of inverse moment estimators for sufficient dimension reduction," Computational Statistics & Data Analysis, Elsevier, vol. 161(C).

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