Convergence of estimators in the polynomial measurement error model
Authors:
A. G. Kukush and Ya. V. Tsaregorodtsev
Translated by:
S. Kvasko
Journal:
Theor. Probability and Math. Statist. 92 (2016), 81-91
MSC (2010):
Primary 62F12, 62J02
DOI:
https://doi.org/10.1090/tpms/984
Published electronically:
August 10, 2016
MathSciNet review:
3553428
Full-text PDF Free Access
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Additional Information
Abstract: A polynomial measurement error model is considered. The variance of errors in the regressor variable and the covariance between errors in the regressor variable and errors of the response variable are assumed to be known. The adjusted least squares estimator of regression parameters adopts the ordinary least squares estimator to the errors presented in the regressor. Conditions for the strong consistency of the estimator are found. These conditions are weaker as compared to those by Cheng and Schneeweiss (1998) [Journal of the Royal Statistical Society B, no. 1, 189–199]. Sufficient conditions for the asymptotic normality of the estimator are also found.
References
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- Alexander Kukush and Erich Otto Maschke, The efficiency of adjusted least squares in the linear functional relationship, J. Multivariate Anal. 87 (2003), no. 2, 261–274. MR 2016938, DOI 10.1016/S0047-259X(03)00048-4
- Alexander Kukush, Andrii Malenko, and Hans Schneeweiss, Optimality of the quasi-score estimator in a mean-variance model with applications to measurement error models, J. Statist. Plann. Inference 139 (2009), no. 10, 3461–3472. MR 2549095, DOI 10.1016/j.jspi.2009.03.022
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References
- P. Billingsley, Convergence of Probability Measures, John Wiley & Sons, Inc., New York–London–Sydney, 1968. MR 0233396
- V. S. Korolyuk, N. I. Portenko, A. V. Skorokhod, and A. F. Turbin, A Manual on Probability Theory and Mathematical Statistics, second edition, “Nauka”, Moscow, 1985. (Russian) MR 828455
- Valentin V. Petrov, Limit theorems of probability theory, Sequences of independent random variables, Oxford Studies in Probability, vol. 4, Oxford Science Publications. The Clarendon Press, Oxford University Press, New York, 1995. MR 1353441
- C.-L. Cheng and H. Schneeweiss, Polynomial regression with errors in the variables, J. Royal Statist. Soc. B 60 (1998), no. 1, 189–199. MR 1625632
- C.-L. Cheng, H. Schneeweiss, and M. Thamerus, A small sample estimator for a polynomial regression with errors in the variables, J. Royal Statist. Soc. 62 (2000), no. 4, 699–709. MR 1796286
- C.-L. Cheng and A. Kukush, Goodness-of-fit test in a polynomial errors-in-variables model, Ukr. Mat. Zh. 56 (2004), no. 4, 527–543; English transl. in Ukrain. Math. J. 56 (2004), no. 4, 641–661. MR 2105906
- P. Hall and Y. Ma, Testing the suitability of polynomial models in errors-in-variables problems, Ann. Statist. 35 (2007), no. 6, 2620–2638. MR 2382660
- A. Kukush and E.-O. Maschke, The efficiency of adjusted least squares in the linear functional relationship, J. Multivariate Anal. 87 (2003), no. 2, 261–274. MR 2016938
- A. Kukush, A. Malenko, and H. Schneeweiss, Optimality of the quasi-score estimator in a mean-variance model with applications to measurement error models, J. Stat. Plann. Inference 139 (2009), no. 10, 3461–3472. MR 2549095
- H. Schneeweiss and A. Kukush, Comparing the efficiency of structural and functional methods in measurement error models, Teor. Ĭmovir. Mat. Stat. 80 (2009), 117–127; English transl. in Theory Probab. Math. Statist. 80 (2010), 131–142. MR 2541958
- I. O. Sen’ko, Consistency of an adjusted least-squares estimator in a vector linear model with measurement errors, Ukr. Mat. Zh. 64 (2012), no. 11, 1536–1546; English transl. in Ukrain. Math. J. 64 (2013), no. 11, 1739–1751. MR 3104843
- I. O. Sen’ko, The asymptotic normality of an adjusted least squares estimator in a multivariate vector errors-in-variables regression model, Teor. Ĭmovir. Mat. Stat. 88 (2013), 157–170; English transl. in Theory Probab. Math. Statist. 88 (2014), 175–190. MR 3112643
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Additional Information
A. G. Kukush
Affiliation:
Department of Mathematical Analysis, Faculty for Mechanics and Mathematics, National Taras Shevchenko University, Academician Glushkov Avenue, 6, Kyiv 03127, Ukraine
Email:
alexander_kukush@univ.kiev.ua
Ya. V. Tsaregorodtsev
Affiliation:
Department of Mathematical Analysis, Faculty for Mechanics and Mathematics, National Taras Shevchenko University, Academician Glushkov Avenue, 6, Kyiv 03127, Ukraine
Email:
777Tsar777@mail.ru
Keywords:
Asymptotic normality,
adjusted least squares estimator,
consistency of estimators,
measurement error model,
modification of estimators for small samples,
polynomial regression
Received by editor(s):
December 23, 2014
Published electronically:
August 10, 2016
Article copyright:
© Copyright 2016
American Mathematical Society
