Consistency of the orthogonal regression estimator in an implicit linear model with errors in variables
Authors:
O. O. Dashkov and A. G. Kukush
Translated by:
S. V. Kvasko
Journal:
Theor. Probability and Math. Statist. 97 (2018), 45-55
MSC (2010):
Primary 62J05; Secondary 62H12, 65F20
DOI:
https://doi.org/10.1090/tpms/1047
Published electronically:
February 21, 2019
MathSciNet review:
3745998
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Additional Information
Abstract: An implicit linear regression model with errors in variables is studied for which the true points belong to a certain hyperplane in Euclidean space and the joint covariance matrix of errors is proportional to the unit matrix. The orthogonal regression estimator for this hyperplane is considered. Some sufficient conditions for the consistency as well as for the strong consistency are given. Some applications to the explicit multiple regression model with a free term and errors in variables are shown.
References
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- 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
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- 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
- S. V. Masiuk, A. G. Kukush, S. V. Shklyar, M. I. Chepurny, and I. A. Likhtarov, Radiation risk estimation, Translated from the 2015 Ukrainian edition, De Gruyter Series in Mathematics and Life Sciences, vol. 5, De Gruyter, Berlin, 2017. Based on measurement error models. MR 3726857
- S. V. Shklyar, Conditions for the consistency of the total least squares estimator in an errors-in-variables linear regression model, Teor. Ĭmovīr. Mat. Stat. 83 (2010), 148–162 (Ukrainian, with English, Russian and Ukrainian summaries); English transl., Theory Probab. Math. Statist. 83 (2011), 175–190. MR 2768857, DOI https://doi.org/10.1090/S0094-9000-2012-00850-8
References
- W. Fuller, Measurement Error Models, Wiley, New York, 1987. MR 898653
- P. Gallo, Consistency of regression estimates when some variables are subject to errors, Comm. Statist. Theory Methods 11 (1982), no. 9, 973–983. MR 655466
- L. J. Gleser, Estimation in multivariate “errors-in-variables” regression model: large sample results, Ann. Statist. 9 (1981), no. 1, 24–44. MR 600530
- A. I. Kostrikin and Yu. I. Manin, Linear Algebra and Geometry, Series in Algebra, Logic and Applications, vol. 1, CRC Press, 1989. MR 1057342
- A. Kukush and S. Van Huffel, Consistency of elementwise-weighted total lest squares estimator in a mulivariate errors-in-variables model $AX=B$, Metrika 59 (2004), no. 1, 75–97. MR 2043433
- S. V. Masiuk, A. G. Kukush, S. V. Shklyar, M. I. Chepurny, I. A. Likhtarov, Radiation Risk Estimation: Based on measurement error models, 2nd ed., De Gruyter Ser. Math. Life Sci., vol. 5, De Gruyter, 2017. MR 3726857
- S. V. Shklyar, Conditions for the consistency of the total least squares estimator in an errors-in-variables linear regression model, Theory Probab. Math. Statist. 83 (2011), 175–190. MR 2768857
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Additional Information
O. O. Dashkov
Affiliation:
Department of Mathematical Analysis, Faculty for Mechanics and Mathematics, Kyiv Taras Shevchenko National University, Volodymyrs’ka Street, 64/13, Kyiv 01601, Ukraine
Email:
oodashkov@gmail.com
A. G. Kukush
Affiliation:
Department of Mathematical Analysis, Faculty for Mechanics and Mathematics, Kyiv Taras Shevchenko National University, Volodymyrs’ka Street, 64/13, Kyiv 01601, Ukraine
Email:
alexander_kukush@univ.kiev.ua
Keywords:
Consistency of an estimator,
multiple regression model with errors in variables,
implicit linear regression model,
orthogonal regression model,
total least squares estimator
Received by editor(s):
October 4, 2017
Published electronically:
February 21, 2019
Article copyright:
© Copyright 2019
American Mathematical Society
