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Nonlinear models with measurement errors subject to single-indexed distortion

Author

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  • Zhang, Jun
  • Zhu, Li-Xing
  • Liang, Hua

Abstract

We study nonlinear regression models whose both response and predictors are measured with errors and distorted as single-index models of some observable confounding variables, and propose a multicovariate-adjusted procedure. We first examine the relationship between the observed primary variables (observed response and observed predictors) and the confounding variables by appropriately estimating the single index. We then develop a semiparametric profile nonlinear least square estimation procedure for the parameters of interest after we calibrate the error-prone response and predictors. Asymptotic properties of the proposed estimators are established. To avoid estimating the asymptotic covariance matrix that contains the infinite-dimensional nuisance distorting functions and the single index, and to improve the accuracy of the proposed estimation, we also propose an empirical likelihood-based statistic, which is shown to be asymptotically chi-squared. A simulation study is conducted to evaluate the performance of the proposed methods and a real dataset is analyzed as an illustration.

Suggested Citation

  • Zhang, Jun & Zhu, Li-Xing & Liang, Hua, 2012. "Nonlinear models with measurement errors subject to single-indexed distortion," Journal of Multivariate Analysis, Elsevier, vol. 112(C), pages 1-23.
    • Handle: RePEc:eee:jmvana:v:112:y:2012:i:c:p:1-23
    DOI: 10.1016/j.jmva.2012.05.012
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    References listed on IDEAS

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    1. Lixing Zhu & Liugen Xue, 2006. "Empirical likelihood confidence regions in a partially linear single‐index model," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 68(3), pages 549-570, June.
    2. Yu Y. & Ruppert D., 2002. "Penalized Spline Estimation for Partially Linear Single-Index Models," Journal of the American Statistical Association, American Statistical Association, vol. 97, pages 1042-1054, December.
    3. Powell, James L & Stock, James H & Stoker, Thomas M, 1989. "Semiparametric Estimation of Index Coefficients," Econometrica, Econometric Society, vol. 57(6), pages 1403-1430, November.
    4. Zhang, Jun & Zhu, Li-Ping & Zhu, Li-Xing, 2012. "On a dimension reduction regression with covariate adjustment," Journal of Multivariate Analysis, Elsevier, vol. 104(1), pages 39-55, February.
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    Citations

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    Cited by:

    1. Xie, Chuanlong & Zhu, Lixing, 2019. "A goodness-of-fit test for variable-adjusted models," Computational Statistics & Data Analysis, Elsevier, vol. 138(C), pages 27-48.
    2. Zhao, Jingxin & Xie, Chuanlong, 2018. "A nonparametric test for covariate-adjusted models," Statistics & Probability Letters, Elsevier, vol. 133(C), pages 65-70.
    3. Zhang, Jun & Li, Gaorong & Feng, Zhenghui, 2015. "Checking the adequacy for a distortion errors-in-variables parametric regression model," Computational Statistics & Data Analysis, Elsevier, vol. 83(C), pages 52-64.
    4. 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.
    5. Yingli Pan & Zhan Liu & Guangyu Song, 2021. "Outlier detection under a covariate-adjusted exponential regression model with censored data," Computational Statistics, Springer, vol. 36(2), pages 961-976, June.
    6. Dai, Shuang & Huang, Zhensheng, 2020. "Nonparametric inference for covariate-adjusted model," Statistics & Probability Letters, Elsevier, vol. 162(C).

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