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On a dimension reduction regression with covariate adjustment

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  • Zhang, Jun
  • Zhu, Li-Ping
  • Zhu, Li-Xing
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    Abstract

    In this paper, we consider a semiparametric modeling with multi-indices when neither the response nor the predictors can be directly observed and there are distortions from some multiplicative factors. In contrast to the existing methods in which the response distortion deteriorates estimation efficacy even for a simple linear model, the dimension reduction technique presented in this paper interestingly does not have to account for distortion of the response variable. The observed response can be used directly whether distortion is present or not. The resulting dimension reduction estimators are shown to be consistent and asymptotically normal. The results can be employed to test whether the central dimension reduction subspace has been estimated appropriately and whether the components in the basis directions in the space are significant. Thus, the method provides an alternative for determining the structural dimension of the subspace and for variable selection. A simulation study is carried out to assess the performance of the proposed method. The analysis of a real dataset demonstrates the potential usefulness of distortion removal.

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    Bibliographic Info

    Article provided by Elsevier in its journal Journal of Multivariate Analysis.

    Volume (Year): 104 (2012)
    Issue (Month): 1 (February)
    Pages: 39-55

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    Handle: RePEc:eee:jmvana:v:104:y:2012:i:1:p:39-55

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    Related research

    Keywords: Central subspace Covariate-adjusted regression Dimension reduction;

    References

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    Please report citation or reference errors to , or , if you are the registered author of the cited work, log in to your RePEc Author Service profile, click on "citations" and make appropriate adjustments.:
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    1. Ye Z. & Weiss R.E., 2003. "Using the Bootstrap to Select One of a New Class of Dimension Reduction Methods," Journal of the American Statistical Association, American Statistical Association, vol. 98, pages 968-979, January.
    2. Liping Zhu & Tao Wang & Lixing Zhu & Louis Ferré, 2010. "Sufficient dimension reduction through discretization-expectation estimation," Biometrika, Biometrika Trust, vol. 97(2), pages 295-304.
    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. DAMLA ŞENTÜRK & HANS-GEORG MÜLLER, 2005. "Covariate Adjusted Correlation Analysis via Varying Coefficient Models," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics & Finnish Statistical Society & Norwegian Statistical Association & Swedish Statistical Association, vol. 32(3), pages 365-383.
    5. Zhu, Li-Ping & Zhu, Li-Xing & Feng, Zheng-Hui, 2010. "Dimension Reduction in Regressions Through Cumulative Slicing Estimation," Journal of the American Statistical Association, American Statistical Association, vol. 105(492), pages 1455-1466.
    6. Heng-Hui Lue, 2004. "Principal Hessian Directions for regression with measurement error," Biometrika, Biometrika Trust, vol. 91(2), pages 409-423, June.
    7. Harrison, David Jr. & Rubinfeld, Daniel L., 1978. "Hedonic housing prices and the demand for clean air," Journal of Environmental Economics and Management, Elsevier, vol. 5(1), pages 81-102, March.
    8. Damla Şentürk & Hans-Georg Müller, 2009. "Covariate-adjusted generalized linear models," Biometrika, Biometrika Trust, vol. 96(2), pages 357-370.
    9. J. Fan & J.-T. Zhang, 2000. "Two-step estimation of functional linear models with applications to longitudinal data," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 62(2), pages 303-322.
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    Cited by:
    1. Zhang, Jun & Feng, Zhenghui & Zhou, Bu, 2014. "A revisit to correlation analysis for distortion measurement error data," Journal of Multivariate Analysis, Elsevier, vol. 124(C), pages 116-129.

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