Conditions for the consistency of the total least squares estimator in an errors-in-variables linear regression model
Author:
S. V. Shklyar
Translated by:
N. Semenov
Journal:
Theor. Probability and Math. Statist. 83 (2011), 175-190
MSC (2010):
Primary 62H12, 62J05; Secondary 62F10, 62F12, 62J10
DOI:
https://doi.org/10.1090/S0094-9000-2012-00850-8
Published electronically:
February 2, 2012
MathSciNet review:
2768857
Full-text PDF Free Access
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Additional Information
Abstract: A homoscedastic errors-in-variables linear regression model is considered. The total least squares estimator is studied. New conditions for the consistency and strong consistency of the total least squares estimator are proposed. These conditions are weaker than those proposed by Kukush and Van Huffel (Metrika 59 (2004), 75–97).
References
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- S. V. Shklyar, An interpolation inequality for moments of sums of random vectors, Teor. Ĭmovīr. Mat. Stat. 63 (2000), 156–162 (Ukrainian, with Ukrainian summary); English transl., Theory Probab. Math. Statist. 63 (2001), 171–177 (2002). MR 1870786
- Wayne A. Fuller, Measurement error models, Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics, John Wiley & Sons, Inc., New York, 1987. MR 898653
- 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
- Leon Jay Gleser, Estimation in a multivariate “errors in variables” regression model: large sample results, Ann. Statist. 9 (1981), no. 1, 24–44. MR 600530
- Wolfgang Härdle, Gerard Kerkyacharian, Dominique Picard, and Alexander Tsybakov, Wavelets, approximation, and statistical applications, Lecture Notes in Statistics, vol. 129, Springer-Verlag, New York, 1998. MR 1618204
- 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
- A. Kukush, I. Markovsky, and S. Van Huffel, Consistency of the structured total least squares estimator in a multivariate errors-in-variables model, J. Statist. Plann. Inference 133 (2005), no. 2, 315–358. MR 2194481, DOI https://doi.org/10.1016/j.jspi.2003.12.020
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- Per-Ȧke Wedin, Perturbation bounds in connection with singular value decomposition, Nordisk Tidskr. Informationsbehandling (BIT) 12 (1972), 99–111. MR 309968, DOI https://doi.org/10.1007/bf01932678
References
- V. V. Petrov, Sums of Independent Random Variables, Nauka, Moscow, 1972; English transl., Springer-Verlag, Berlin–New York–Heidelberg, 1975. MR 0322927 (48:1288)
- S. V. Shklyar, An interpolation inequality for moments of sums of random vectors, Teor. Imovir. Mat. Stat. 63 (2000), 156–162; English transl. in Theory Probab. Math. Statist. 63 (2001), 171–177. MR 1870786 (2002m:60041)
- W. Fuller, Measurement Error Models, Wiley, New York, 1987. MR 898653 (89a:62160)
- P. Gallo, Consistency of regression estimates when some variables are subject to error, Commun. Statist. Theor. Meth. 11 (1982), no. 9, 973–983. MR 655466 (83h:62106)
- L. J. Gleser, Estimation in a multivariate “errors in variables” regression model: Large sample results, Ann. Statist. 9 (1981), 24–44. MR 600530 (82c:62096)
- W. Härdle, G. Kerkyacharian, D. Picard, and A. Tsybakov, Wavelets, Approximation, and Statistical Applications, Lecture Notes in Statistics, vol. 129, Springer, New York, 1998. MR 1618204 (99f:42065)
- A. Kukush and S. Van Huffel, Consistency of element-weighted total least squares estimator in multivariate errors-in-variables model $A X = B$, Metrika 59 (2004), no. 1, 75–97. MR 2043433 (2004m:62129)
- A. Kukush, I. Markovsky, and S. Van Huffel, Consistency of structured total least squares estimator in a multivariate errors-in-variables model, J. Statist. Planning Inference 133 (2005), 315–358. MR 2194481 (2007b:62030)
- G. W. Stewart and J.-G. Sun, Matrix Perturbation Theory, Academic Press, San Diego, 1990. MR 1061154 (92a:65017)
- P.-Å. Wedin, Perturbation bounds in connection with singular value decomposition, BIT 12 (1972), 99–111. MR 0309968 (46:9071)
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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, Academician Glushkov Avenue 2, Kiev 03127, Ukraine
Email:
shklyar@mail.univ.kiev.ua
Keywords:
Linear regression,
errors in variables,
total least squares estimator,
orthogonal regression
Received by editor(s):
November 8, 2010
Published electronically:
February 2, 2012
Article copyright:
© Copyright 2012
American Mathematical Society
