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Dimension reduction for non-elliptically distributed predictors: second-order methods

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  • Yuexiao Dong
  • Bing Li

Abstract

Many classical dimension reduction methods, especially those based on inverse conditional moments, require the predictors to have elliptical distributions, or at least to satisfy a linearity condition. Such conditions, however, are too strong for some applications. Li and Dong (2009) introduced the notion of the central solution space and used it to modify first-order methods, such as sliced inverse regression, so that they no longer rely on these conditions. In this paper we generalize this idea to second-order methods, such as sliced average variance estimation and directional regression. In doing so we demonstrate that the central solution space is a versatile framework: we can use it to modify essentially all inverse conditional moment-based methods to relax the distributional assumption on the predictors. Simulation studies and an application show a substantial improvement of the modified methods over their classical counterparts. Copyright 2010, Oxford University Press.

Suggested Citation

  • Yuexiao Dong & Bing Li, 2010. "Dimension reduction for non-elliptically distributed predictors: second-order methods," Biometrika, Biometrika Trust, vol. 97(2), pages 279-294.
  • Handle: RePEc:oup:biomet:v:97:y:2010:i:2:p:279-294
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    Cited by:

    1. Ming-Yueh Huang & Chin-Tsang Chiang, 2017. "An Effective Semiparametric Estimation Approach for the Sufficient Dimension Reduction Model," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 112(519), pages 1296-1310, July.
    2. Deng, Jianqiu & Yang, Xiaojie & Wang, Qihua, 2022. "Surrogate space based dimension reduction for nonignorable nonresponse," Computational Statistics & Data Analysis, Elsevier, vol. 168(C).
    3. Feng, Zhenghui & Wang, Tao & Zhu, Lixing, 2014. "Transformation-based estimation," Computational Statistics & Data Analysis, Elsevier, vol. 78(C), pages 186-205.
    4. Kangning Wang & Lu Lin, 2017. "Robust and efficient direction identification for groupwise additive multiple-index models and its applications," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 26(1), pages 22-45, March.
    5. Lu Li & Kai Tan & Xuerong Meggie Wen & Zhou Yu, 2023. "Variable-dependent partial dimension reduction," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 32(2), pages 521-541, June.
    6. Ding, Shanshan & Cook, R. Dennis, 2015. "Tensor sliced inverse regression," Journal of Multivariate Analysis, Elsevier, vol. 133(C), pages 216-231.
    7. Wang, Qin & Xue, Yuan, 2021. "An ensemble of inverse moment estimators for sufficient dimension reduction," Computational Statistics & Data Analysis, Elsevier, vol. 161(C).
    8. Dong, Yuexiao & Yu, Zhou, 2012. "Dimension reduction for the conditional kth moment via central solution space," Journal of Multivariate Analysis, Elsevier, vol. 112(C), pages 207-218.
    9. Wenjuan Li & Wenying Wang & Jingsi Chen & Weidong Rao, 2023. "Aggregate Kernel Inverse Regression Estimation," Mathematics, MDPI, vol. 11(12), pages 1-10, June.
    10. Jianxuan Liu & Yanyuan Ma & Lan Wang, 2018. "An alternative robust estimator of average treatment effect in causal inference," Biometrics, The International Biometric Society, vol. 74(3), pages 910-923, September.
    11. Zhou Yu & Yuexiao Dong & Li-Xing Zhu, 2016. "Trace Pursuit: A General Framework for Model-Free Variable Selection," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 111(514), pages 813-821, April.
    12. Chen, Fei & Shi, Lei & Zhu, Xuehu & Zhu, Lixing, 2018. "Generalized principal Hessian directions for mixture multivariate skew elliptical distributions," Journal of Multivariate Analysis, Elsevier, vol. 168(C), pages 142-159.
    13. Dong, Yuexiao & Yu, Zhou & Zhu, Liping, 2015. "Robust inverse regression for dimension reduction," Journal of Multivariate Analysis, Elsevier, vol. 134(C), pages 71-81.
    14. Xie, Chuanlong & Zhu, Lixing, 2020. "Generalized kernel-based inverse regression methods for sufficient dimension reduction," Computational Statistics & Data Analysis, Elsevier, vol. 150(C).
    15. Eliana Christou, 2020. "Robust dimension reduction using sliced inverse median regression," Statistical Papers, Springer, vol. 61(5), pages 1799-1818, October.
    16. Dong, Yuexiao & Xia, Qi & Tang, Cheng Yong & Li, Zeda, 2018. "On sufficient dimension reduction with missing responses through estimating equations," Computational Statistics & Data Analysis, Elsevier, vol. 126(C), pages 67-77.
    17. 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.
    18. Yu, Zhou & Dong, Yuexiao & Huang, Mian, 2014. "General directional regression," Journal of Multivariate Analysis, Elsevier, vol. 124(C), pages 94-104.

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