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Optimal transformation: A new approach for covering the central subspace

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  • Portier, François
  • Delyon, Bernard

Abstract

This paper studies a general family of methods for sufficient dimension reduction (SDR) called the test function (TF), based on the introduction of a nonlinear transformation of the response. By considering order 1 and 2 conditional moments of the predictors given the response, we distinguish two classes of methods. The optimal members of each class are calculated with respect to the asymptotic mean squared error between the central subspace (CS) and its estimate. Moreover the theoretical background of TF is developed under weaker conditions than the existing methods. Accordingly, simulations confirm that the resulting methods are highly accurate.

Suggested Citation

  • Portier, François & Delyon, Bernard, 2013. "Optimal transformation: A new approach for covering the central subspace," Journal of Multivariate Analysis, Elsevier, vol. 115(C), pages 84-107.
    • Handle: RePEc:eee:jmvana:v:115:y:2013:i:c:p:84-107
    DOI: 10.1016/j.jmva.2012.09.001
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    References listed on IDEAS

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    1. Eaton, Morris L., 1986. "A characterization of spherical distributions," Journal of Multivariate Analysis, Elsevier, vol. 20(2), pages 272-276, December.
    2. Bura, E. & Yang, J., 2011. "Dimension estimation in sufficient dimension reduction: A unifying approach," Journal of Multivariate Analysis, Elsevier, vol. 102(1), pages 130-142, January.
    3. Li, Bing & Wang, Shaoli, 2007. "On Directional Regression for Dimension Reduction," Journal of the American Statistical Association, American Statistical Association, vol. 102, pages 997-1008, September.
    4. Yingcun Xia & Howell Tong & W. K. Li & Li‐Xing Zhu, 2002. "An adaptive estimation of dimension reduction space," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 64(3), pages 363-410, August.
    5. Xiangrong Yin & R. Dennis Cook, 2002. "Dimension reduction for the conditional kth moment in regression," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 64(2), pages 159-175, May.
    6. Cook, R. Dennis & Ni, Liqiang, 2005. "Sufficient Dimension Reduction via Inverse Regression: A Minimum Discrepancy Approach," Journal of the American Statistical Association, American Statistical Association, vol. 100, pages 410-428, June.
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    Cited by:

    1. François Portier, 2016. "An Empirical Process View of Inverse Regression," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 43(3), pages 827-844, September.
    2. Paris, Quentin, 2014. "Minimax adaptive dimension reduction for regression," Journal of Multivariate Analysis, Elsevier, vol. 128(C), pages 186-202.

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