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Theory of Probability and Mathematical Statistics

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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
Original publication: Teoriya Imovirnostei ta Matematichna Statistika, tom 76 (2007).
Journal: Theor. Probability and Math. Statist. 76 (2008), 181-197
MSC (2000): Primary 62J02; Secondary 62F10, 62F12, 62J10
Published electronically: July 17, 2008
MathSciNet review: 2368750
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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 [Enhancements On Off] (What's this?)

  • 1. C.-L. Cheng and J. Van Ness, Statistical Regression with Measurement Error, Arnold, London, 1999. MR 1719513 (2001k:62001)
  • 2. C.-L. Cheng and H. Schneeweiss, Polynomial regression with measurement errors, J. Roy. Statist. Soc. Ser. B 60 (1998), 189-199. MR 1625632
  • 3. 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
  • 4. 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)
  • 5. Zh. Zhang, Parameter estimation techniques, Image & Vision Computing J. 15 (1997), no. 1, 59-76.
  • 6. 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)
  • 7. 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)
  • 8. 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
  • 9. G. Stewart and J. Sun, Matrix Perturbation Theory, Academic Press, London, 1990. MR 1061154 (92a:65017)
  • 10. 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)
  • 11. 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

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

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