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Multiplicative regression models with distortion measurement errors

Author

Listed:
  • Jun Zhang

    (Shenzhen University)

  • Junpeng Zhu

    (Shenzhen University)

  • Yan Zhou

    (Shenzhen University)

  • Xia Cui

    (Guangzhou University)

  • Tao Lu

    (University of Nevada)

Abstract

This paper considers estimation and variable selection for multiplicative linear regression models when neither the response variable nor the covariates can be directly observed, but are distorted by unknown functions of a commonly observable confounding variable. After taking logarithmic transformation on the response variable, we propose two estimation methods for the parameter. One is the least squares estimator, the second one is the moment-based estimator linked with varying coefficient models. The third one is the least product relative error estimator without logarithmic transformation. For the hypothesis testing of parametric components, restricted estimators under the null hypothesis and test statistics are proposed. The asymptotic properties for the estimators and test statistics are established. A bootstrap procedure is proposed to calculate critical values. A smoothly clipped absolute deviation penalty is employed to select the relevant variables. The resulting penalized estimators are shown to be asymptotically normal and have the oracle property. Simulation studies demonstrate the performance of the proposed procedure and a real example is analyzed to illustrate its practical usage.

Suggested Citation

  • Jun Zhang & Junpeng Zhu & Yan Zhou & Xia Cui & Tao Lu, 2020. "Multiplicative regression models with distortion measurement errors," Statistical Papers, Springer, vol. 61(5), pages 2031-2057, October.
    • Handle: RePEc:spr:stpapr:v:61:y:2020:i:5:d:10.1007_s00362-018-1020-2
    DOI: 10.1007/s00362-018-1020-2
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    References listed on IDEAS

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    1. Chuan-hua Wei & Chunling Liu, 2012. "Statistical inference on semi-parametric partial linear additive models," Journal of Nonparametric Statistics, Taylor & Francis Journals, vol. 24(4), pages 809-823, December.
    2. Chen, Kani & Lin, Yuanyuan & Wang, Zhanfeng & Ying, Zhiliang, 2016. "Least product relative error estimation," Journal of Multivariate Analysis, Elsevier, vol. 144(C), pages 91-98.
    3. Danh Nguyen & Damla şentürk & Raymond Carroll, 2008. "Covariate-adjusted linear mixed effects model with an application to longitudinal data," Journal of Nonparametric Statistics, Taylor & Francis Journals, vol. 20(6), pages 459-481.
    4. Pötscher, Benedikt M. & Leeb, Hannes, 2009. "On the distribution of penalized maximum likelihood estimators: The LASSO, SCAD, and thresholding," Journal of Multivariate Analysis, Elsevier, vol. 100(9), pages 2065-2082, October.
    5. Fan J. & Li R., 2001. "Variable Selection via Nonconcave Penalized Likelihood and its Oracle Properties," Journal of the American Statistical Association, American Statistical Association, vol. 96, pages 1348-1360, December.
    6. Chen, Kani & Guo, Shaojun & Lin, Yuanyuan & Ying, Zhiliang, 2010. "Least Absolute Relative Error Estimation," Journal of the American Statistical Association, American Statistical Association, vol. 105(491), pages 1104-1112.
    7. Wang, Hansheng & Xia, Yingcun, 2009. "Shrinkage Estimation of the Varying Coefficient Model," Journal of the American Statistical Association, American Statistical Association, vol. 104(486), pages 747-757.
    8. Zhao, Jingxin & Xie, Chuanlong, 2018. "A nonparametric test for covariate-adjusted models," Statistics & Probability Letters, Elsevier, vol. 133(C), pages 65-70.
    9. Senturk, Damla & Nguyen, Danh V., 2006. "Estimation in covariate-adjusted regression," Computational Statistics & Data Analysis, Elsevier, vol. 50(11), pages 3294-3310, July.
    10. Escanciano, J. Carlos, 2006. "A Consistent Diagnostic Test For Regression Models Using Projections," Econometric Theory, Cambridge University Press, vol. 22(6), pages 1030-1051, December.
    11. Chuanhua Wei & Qihua Wang, 2012. "Statistical inference on restricted partially linear additive errors-in-variables models," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 21(4), pages 757-774, December.
    12. Feng Li & Yiqiang Lu, 2018. "Lasso-type estimation for covariate-adjusted linear model," Journal of Applied Statistics, Taylor & Francis Journals, vol. 45(1), pages 26-42, January.
    13. Liang, Hua & Li, Runze, 2009. "Variable Selection for Partially Linear Models With Measurement Errors," Journal of the American Statistical Association, American Statistical Association, vol. 104(485), pages 234-248.
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    Cited by:

    1. Jun Zhang & Bingqing Lin & Yiping Yang, 2022. "Maximum nonparametric kernel likelihood estimation for multiplicative linear regression models," Statistical Papers, Springer, vol. 63(3), pages 885-918, June.
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    3. Huilan Liu & Xiawei Zhang & Huaiqing Hu & Junjie Ma, 2024. "Analysis of the positive response data with the varying coefficient partially nonlinear multiplicative model," Statistical Papers, Springer, vol. 65(5), pages 3063-3092, July.
    4. Yinjun Chen & Hao Ming & Hu Yang, 2024. "Efficient variable selection for high-dimensional multiplicative models: a novel LPRE-based approach," Statistical Papers, Springer, vol. 65(6), pages 3713-3737, August.

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