The uniqueness of the quasi-likelihood estimator in the Poisson model with an error in the regressor
Author:
S. V. Shklyar
Translated by:
S. Kvasko
Journal:
Theor. Probability and Math. Statist. 88 (2014), 203-216
MSC (2010):
Primary 62J12; Secondary 62H10
DOI:
https://doi.org/10.1090/S0094-9000-2014-00928-X
Published electronically:
July 24, 2014
MathSciNet review:
3112645
Full-text PDF Free Access
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Abstract: The Poisson regression Berkson type model with a Gaussian error in the regressor is studied. Simple score and quasi-likelihood estimators of the regression parameters are considered. Sufficient conditions for the strong consistency of the estimators and sufficient conditions for the uniqueness of a solution of estimating equations are found. The proof of the uniqueness does not require that the parameter set be bounded.
References
- Raymond J. Carroll, David Ruppert, Leonard A. Stefanski, and Ciprian M. Crainiceanu, Measurement error in nonlinear models, 2nd ed., Monographs on Statistics and Applied Probability, vol. 105, Chapman & Hall/CRC, Boca Raton, FL, 2006. A modern perspective. MR 2243417
- Alexander Kukush, Hans Schneeweis, and Roland Wolf, Three estimators for the Poisson regression model with measurement errors, Statist. Papers 45 (2004), no. 3, 351–368. MR 2064718, DOI https://doi.org/10.1007/BF02777577
- A. Kukush and H. Schneeweiss, Comparing different estimators in a nonlinear measurement error model. I, Math. Methods Statist. 14 (2005), no. 1, 53–79. MR 2158071
- R. Tyrrell Rockafellar, Convex analysis, Princeton Mathematical Series, No. 28, Princeton University Press, Princeton, N.J., 1970. MR 0274683
- H. Schneeweiss, The polynomial and the Poisson measurement error models: some further results on quasi score and corrected score estimation, SFB 386, Discussion Paper 446, LMU Munich, 2005.
- S. V. Shklyar, Asymptotic properties of estimators of parameters of nonlinear errors-in-variables regression models, Candidate Dissertation, Kyiv National Taras Shevchenko University, Kyiv, 2008. (Ukrainian)
- Markus Thamerus, Different nonlinear regression models with incorrectly observed covariates, Econometrics in theory and practice, Physica, Heidelberg, 1998, pp. 31–44. MR 1655411
- R. W. M. Wedderburn, Quasi-likelihood functions, generalized linear models, and the Gauss-Newton method, Biometrika 61 (1974), 439–447. MR 375592, DOI https://doi.org/10.1093/biomet/61.3.439
References
- R. J. Carroll, D. Ruppert, L. A. Stefanski, and C. M. Crainiceanu, Measurement Error in Nonlinear Models, Second edition, Chapman & Hall, Boca Raton, 2006. MR 2243417 (2007e:62004)
- A. Kukush, H. Schneeweiss, and R. Wolf, Three estimators for the Poisson regression model with measurement errors, Statistical Papers 45 (2000), no. 3, 351–568. MR 2064718
- A. Kukush and H. Schneeweiss, Comparing different estimators in a nonlinear measurement error model, Part 1, Mathematical Methods of Statistics 14 (2005), no. 1, 53–79. MR 2158071 (2006j:62068a)
- R. Rokafellar, Convex Analysis, Princeton University Press, Princeton, New Jersey, 1970. MR 0274683 (43:445)
- H. Schneeweiss, The polynomial and the Poisson measurement error models: some further results on quasi score and corrected score estimation, SFB 386, Discussion Paper 446, LMU Munich, 2005.
- S. V. Shklyar, Asymptotic properties of estimators of parameters of nonlinear errors-in-variables regression models, Candidate Dissertation, Kyiv National Taras Shevchenko University, Kyiv, 2008. (Ukrainian)
- M. Thamerus, Different nonlinear regression models with incorrectly observed covariates, Econometrics in Theory and Practice (R. Galata and H. Kuechenhoff, eds.), Physica-Verlag, Heidelberg, 1998, pp. 31–44. MR 1655411
- R. W. M. Wedderburn, Quasi-likelihood functions, generalized linear models, and the Gauss–Newton method, Biometrika 61 (1974), no. 3, 439–447. MR 0375592 (51:11783)
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Additional Information
S. V. Shklyar
Affiliation:
Department of Probability Theory, Statistics, and Actuarial Mathematics, Faculty for Mechanics and Mathematics, National Taras Shevchenko University, Volodymyrs’ka Street, 64, Kyiv 01601, Ukraine
Email:
shklyar@mail.univ.kiev.ua
Keywords:
Regression errors-in-variables model,
Poisson regression,
Berkson type model
Received by editor(s):
October 12, 2012
Published electronically:
July 24, 2014
Article copyright:
© Copyright 2014
American Mathematical Society
