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Likelihood Inference in the Errors-in-Variables Model

Author

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  • Murphy, S. A.
  • Van Der Vaart, A. W.

Abstract

We consider estimation and confidence regions for the parameters[alpha]and[beta]based on the observations (X1, Y1), ..., (Xn, Yn) in the errors-in-variables modelXi=Zi+eiandYi=[alpha]+[beta]Zi+fifor normal errorseiandfiof which the covariance matrix is known up to a constant. We study the asymptotic performance of the estimators defined as the maximum likelihood estimator under the assumption thatZ1, ..., Znis a random sample from a completely unknown distribution. These estimators are shown to be asymptotically efficient in the semi-parametric sense if this assumption is valid. These estimators are shown to be asymptotically normal even in the case thatZ1, Z2, ... are arbitrary constants satisfying a moment condition. Similarly we study the confidence regions obtained from the likelihood ratio statistic for the mixture model and show that these are asymptotically consistent both in the mixture case and in the case thatZ1, Z2, ... are arbitrary constants.

Suggested Citation

  • Murphy, S. A. & Van Der Vaart, A. W., 1996. "Likelihood Inference in the Errors-in-Variables Model," Journal of Multivariate Analysis, Elsevier, vol. 59(1), pages 81-108, October.
    • Handle: RePEc:eee:jmvana:v:59:y:1996:i:1:p:81-108
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    Cited by:

    1. Yingyao Hu & Geert Ridder, 2012. "Estimation of nonlinear models with mismeasured regressors using marginal information," Journal of Applied Econometrics, John Wiley & Sons, Ltd., vol. 27(3), pages 347-385, April.
    2. Xiaohong Chen & Yingyao Hu & Arthur Lewbel, 2007. "Nonparametric identification and estimation of nonclassical errors-in-variables models without additional information," CeMMAP working papers CWP18/07, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.
    3. Jingjing Wu & Rohana J. Karunamuni, 2018. "Efficient and robust tests for semiparametric models," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 70(4), pages 761-788, August.
    4. Wu, Jingjing & Karunamuni, Rohana J., 2012. "Efficient Hellinger distance estimates for semiparametric models," Journal of Multivariate Analysis, Elsevier, vol. 107(C), pages 1-23.
    5. Marie-Luce Taupin, 2010. "Comment on identification and estimation of nonlinear models using two samples with nonclassical measurement errors," Journal of Nonparametric Statistics, Taylor & Francis Journals, vol. 22(4), pages 409-414.

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