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A Semiparametric Approach to Dimension Reduction

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  • Yanyuan Ma
  • Liping Zhu

Abstract

We provide a novel and completely different approach to dimension-reduction problems from the existing literature. We cast the dimension-reduction problem in a semiparametric estimation framework and derive estimating equations. Viewing this problem from the new angle allows us to derive a rich class of estimators, and obtain the classical dimension reduction techniques as special cases in this class. The semiparametric approach also reveals that in the inverse regression context while keeping the estimation structure intact, the common assumption of linearity and/or constant variance on the covariates can be removed at the cost of performing additional nonparametric regression. The semiparametric estimators without these common assumptions are illustrated through simulation studies and a real data example. This article has online supplementary material.

Suggested Citation

  • Yanyuan Ma & Liping Zhu, 2012. "A Semiparametric Approach to Dimension Reduction," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 107(497), pages 168-179, March.
  • Handle: RePEc:taf:jnlasa:v:107:y:2012:i:497:p:168-179
    DOI: 10.1080/01621459.2011.646925
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    1. repec:bla:scjsta:v:44:y:2017:i:1:p:112-129 is not listed on IDEAS
    2. repec:eee:stapro:v:126:y:2017:i:c:p:108-113 is not listed on IDEAS
    3. repec:spr:sankha:v:80:y:2018:i:1:d:10.1007_s13171-017-0102-x is not listed on IDEAS
    4. Luo, Wei & Cai, Xizhen, 2016. "A new estimator for efficient dimension reduction in regression," Journal of Multivariate Analysis, Elsevier, vol. 145(C), pages 236-249.
    5. Yanyuan Ma, 2015. "Discussion," International Statistical Review, International Statistical Institute, vol. 83(2), pages 207-211, August.
    6. Feng, Zhenghui & Wang, Tao & Zhu, Lixing, 2014. "Transformation-based estimation," Computational Statistics & Data Analysis, Elsevier, vol. 78(C), pages 186-205.
    7. repec:bla:biomet:v:72:y:2016:i:4:p:1275-1284 is not listed on IDEAS
    8. repec:oup:biomet:v:104:y:2017:i:3:p:583-596. is not listed on IDEAS
    9. Iaci, Ross & Yin, Xiangrong & Zhu, Lixing, 2016. "The Dual Central Subspaces in dimension reduction," Journal of Multivariate Analysis, Elsevier, vol. 145(C), pages 178-189.
    10. Yanyuan Ma & Raymond J. Carroll, 2016. "Semiparametric estimation in the secondary analysis of case–control studies," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 78(1), pages 127-151, January.
    11. Dong, Yuexiao & Yu, Zhou & Zhu, Liping, 2015. "Robust inverse regression for dimension reduction," Journal of Multivariate Analysis, Elsevier, vol. 134(C), pages 71-81.
    12. repec:bla:scjsta:v:44:y:2017:i:1:p:1-20 is not listed on IDEAS
    13. Liu, Xuejing & Huo, Lei & Wen, Xuerong Meggie & Paige, Robert, 2017. "A link-free approach for testing common indices for three or more multi-index models," Journal of Multivariate Analysis, Elsevier, vol. 153(C), pages 236-245.
    14. Ding, Shanshan & Cook, R. Dennis, 2015. "Tensor sliced inverse regression," Journal of Multivariate Analysis, Elsevier, vol. 133(C), pages 216-231.
    15. Sheng, Wenhui & Yin, Xiangrong, 2013. "Direction estimation in single-index models via distance covariance," Journal of Multivariate Analysis, Elsevier, vol. 122(C), pages 148-161.
    16. Zhou, Jingke & Xu, Wangli & Zhu, Lixing, 2015. "Robust estimating equation-based sufficient dimension reduction," Journal of Multivariate Analysis, Elsevier, vol. 134(C), pages 99-118.
    17. Zhou, Jingke & Zhu, Lixing, 2016. "Principal minimax support vector machine for sufficient dimension reduction with contaminated data," Computational Statistics & Data Analysis, Elsevier, vol. 94(C), pages 33-48.

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