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Estimation of Partial Linear Error-in-Variables Models with Validation Data

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  • Wang, Qihua

Abstract

Consider the partial linear models of the formY=X[tau][beta]+g(T)+e, where thep-variate explanatoryXis erroneously measured, and bothTand the responseYare measured exactly. LetXbe the surrogate variable forXwith measurement error. Let the primary data set be that containing independent observations on (Y, X, T) and the validation data set be that containing independent observations on (X, X, T), where the exact observations onXmay be obtained by some expensive or difficult procedures for only a small subset of subjects enrolled in the study. In this paper, without specifying any structure equation and the distribution assumption ofXgivenX, a semiparametric method with the primary data is employed to obtain the estimators of[beta]andg(·) based on the least-squares criterion with the help of validation data. The proposed estimators are proved to be strongly consistent. The asymptotic representation and the asymptotic normality of the estimator of[beta]are derived, respectively. The rate of the weak consistency of the estimator ofg(·) is also obtained.

Suggested Citation

  • Wang, Qihua, 1999. "Estimation of Partial Linear Error-in-Variables Models with Validation Data," Journal of Multivariate Analysis, Elsevier, vol. 69(1), pages 30-64, April.
    • Handle: RePEc:eee:jmvana:v:69:y:1999:i:1:p:30-64
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    References listed on IDEAS

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    1. Sepanski, J. H. & Carroll, R. J., 1993. "Semiparametric quasilikelihood and variance function estimation in measurement error models," Journal of Econometrics, Elsevier, vol. 58(1-2), pages 223-256, July.
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    Cited by:

    1. Xu, Hong-Xia & Fan, Guo-Liang & Chen, Zhen-Long, 2017. "Hypothesis tests in partial linear errors-in-variables models with missing response," Statistics & Probability Letters, Elsevier, vol. 126(C), pages 219-229.
    2. Wang, Xuan & Wang, Qihua, 2015. "Semiparametric linear transformation model with differential measurement error and validation sampling," Journal of Multivariate Analysis, Elsevier, vol. 141(C), pages 67-80.
    3. Wang, Qihua, 2003. "Dimension reduction in partly linear error-in-response models with validation data," Journal of Multivariate Analysis, Elsevier, vol. 85(2), pages 234-252, May.
    4. Song, Weixing, 2009. "Lack-of-fit testing in errors-in-variables regression model with validation data," Statistics & Probability Letters, Elsevier, vol. 79(6), pages 765-773, March.
    5. 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.
    6. Wangli Xu & Lixing Zhu, 2015. "Nonparametric check for partial linear errors-in-covariables models with validation data," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 67(4), pages 793-815, August.
    7. Wang, Qihua & Zhang, Riquan, 2009. "Statistical estimation in varying coefficient models with surrogate data and validation sampling," Journal of Multivariate Analysis, Elsevier, vol. 100(10), pages 2389-2405, November.
    8. Wang, Qihua, 2006. "Nonparametric regression function estimation with surrogate data and validation sampling," Journal of Multivariate Analysis, Elsevier, vol. 97(5), pages 1142-1161, May.
    9. Wang, Qihua, 2000. "Estimation of Linear Error-in-Covariables Models with Validation Data Under Random Censorship," Journal of Multivariate Analysis, Elsevier, vol. 74(2), pages 245-266, August.
    10. Qi-Hua Wang, 2009. "Statistical estimation in partial linear models with covariate data missing at random," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 61(1), pages 47-84, March.
    11. Wang, Qihua & Yu, Keming, 2007. "Likelihood-based kernel estimation in semiparametric errors-in-covariables models with validation data," Journal of Multivariate Analysis, Elsevier, vol. 98(3), pages 455-480, March.
    12. Chuanhua Wei & Jin Yang, 2020. "Stochastic restricted estimation in partially linear additive errors-in-variables models," Statistical Papers, Springer, vol. 61(3), pages 1269-1279, June.

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