Consistency of an estimator of the parameters of a polynomial regression with a known variance relation for errors in the measurement of the regressor and the echo
Author:
S. V. Shklyar
Translated by:
N. Semenov
Journal:
Theor. Probability and Math. Statist. 76 (2008), 181-197
MSC (2000):
Primary 62J02; Secondary 62F10, 62F12, 62J10
DOI:
https://doi.org/10.1090/S0094-9000-08-00742-4
Published electronically:
July 17, 2008
MathSciNet review:
2368750
Full-text PDF Free Access
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Additional Information
Abstract: We consider an error-in-variables model for a polynomial regression with Gaussian errors. We assume that the covariance matrix of the measurement errors of the regressor and the echo is known up to a scalar factor. We consider the moment estimator of regression coefficients proposed by Cheng and Schneeweiss. Sufficient conditions for the strong consistency of this estimator are given and the rate of convergence is estimated in this paper.
References
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- Chi-Lun Cheng and Hans Schneeweiss, Polynomial regression with errors in the variables, J. R. Stat. Soc. Ser. B Stat. Methodol. 60 (1998), no. 1, 189–199. MR 1625632, DOI https://doi.org/10.1111/1467-9868.00118
- Chi-Lun Cheng and Hans Schneeweiss, On the polynomial measurement error model, Total least squares and errors-in-variables modeling (Leuven, 2001) Kluwer Acad. Publ., Dordrecht, 2002, pp. 131–143. MR 1952942
- G. S. Repetats′ka, Inconsistency of an orthogonal regression estimator in a vector nonlinear error-variables model, Teor. Ĭmovīr. Mat. Stat. 73 (2005), 146–160 (Ukrainian, with Ukrainian summary); English transl., Theory Probab. Math. Statist. 73 (2006), 163–179. MR 2213850, DOI https://doi.org/10.1090/S0094-9000-07-00690-4
- Zh. Zhang, Parameter estimation techniques, Image & Vision Computing J. 15 (1997), no. 1, 59–76.
- Alexander Kukush, Ivan Markovsky, and Sabine Van Huffel, Consistent estimation in an implicit quadratic measurement error model, Comput. Statist. Data Anal. 47 (2004), no. 1, 123–147. MR 2087933, DOI https://doi.org/10.1016/j.csda.2003.10.022
- Sergiy Shklyar, Alexander Kukush, Ivan Markovsky, and Sabine Van Huffel, On the conic section fitting problem, J. Multivariate Anal. 98 (2007), no. 3, 588–624. MR 2293016, DOI https://doi.org/10.1016/j.jmva.2005.12.003
- Sergiy Shklyar, Hans Schneeweiss, and Alexander Kukush, Quasi score is more efficient than corrected score in a polynomial measurement error model, Metrika 65 (2007), no. 3, 275–295. MR 2299552, DOI https://doi.org/10.1007/s00184-006-0076-5
- G. W. Stewart and Ji Guang Sun, Matrix perturbation theory, Computer Science and Scientific Computing, Academic Press, Inc., Boston, MA, 1990. MR 1061154
- Paul P. Gallo, Consistency of regression estimates when some variables are subject to error, Comm. Statist. A—Theory Methods 11 (1982), no. 9, 973–983. MR 655466, DOI https://doi.org/10.1080/03610928208828287
- Alexander Kukush and Sabine Van Huffel, Consistency of elementwise-weighted total least squares estimator in a multivariate errors-in-variables model $AX=B$, Metrika 59 (2004), no. 1, 75–97. MR 2043433, DOI https://doi.org/10.1007/s001840300272
References
- C.-L. Cheng and J. Van Ness, Statistical Regression with Measurement Error, Arnold, London, 1999. MR 1719513 (2001k:62001)
- C.-L. Cheng and H. Schneeweiss, Polynomial regression with measurement errors, J. Roy. Statist. Soc. Ser. B 60 (1998), 189–199. MR 1625632
- C.-L. Cheng and H. Schneeweiss, On the polynomial measurement error model, Total Least Squares and Error-In-Variables Modelling (S. Van Huffel and Ph. Lemmerling, eds.), Kluwer, Dordrecht, 2002, pp. 131–143. MR 1952942
- G. S. Repetats’ka, Inconsistency of an orthogonal regression estimator in a vector nonlinear errors-in-variables model, Teor. Imovir. Mat. Stat. 73 (2005), 146–160; English transl. in Theory Probab. Math. Statist. 73 (2006), 163–179. MR 2213850 (2007g:62072)
- Zh. Zhang, Parameter estimation techniques, Image & Vision Computing J. 15 (1997), no. 1, 59–76.
- A. Kukush, I. Markovsky, and S. Van Huffel, Consistent estimation in an implicit quadratic measurement error model, Comput. Statist. Data Anal. 47 (2004), no. 1, 123–147. MR 2087933 (2005h:62077)
- S. Shklyar, A. Kukush, I. Markovsky, and S. Van Huffel, On the conic section fitting problem, J. Multivariate Anal. 98 (2007), no. 3, 588–642. MR 2293016 (2008g:62164)
- S. Shklyar, H. Schneeweiss, and A. Kukush, Quasi score is more efficient than corrected score in a polynomial measurement error model, Metrika 65 (2007), no. 3, 275–295. MR 2299552
- G. Stewart and J. Sun, Matrix Perturbation Theory, Academic Press, London, 1990. MR 1061154 (92a:65017)
- P. Gallo, Consistency of regression estimates when some variables are subject to error, Commun. Stat. Theor. Meth. 11 (1982), no. 9, 973–983. MR 655466 (83h:62106)
- A. G. Kukush and S. Van Huffel, Consistency of elementwise-weighted total least squares estimator in a multivariate error-in-variables model $AX=B$, Metrika 59 (2004), no. 1, 75–97. MR 2043433 (2004m:62129)
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Additional Information
S. V. Shklyar
Affiliation:
Department of Probability Theory and Mathematical Statistics, Faculty for Mechanics and Mathematics, National Taras Shevchenko University, Academician Glushkov Avenue 6, Kyiv 03127, Ukraine
Email:
shklyar@mail.univ.kiev.ua
Keywords:
Polynomial regression,
error-in-variables model
Received by editor(s):
February 24, 2006
Published electronically:
July 17, 2008
Article copyright:
© Copyright 2008
American Mathematical Society
