Minimax estimators of parameters of a regression model
Authors:
A. V. Ivanov and I. K. Matsak
Translated by:
S. V. Kvasko
Journal:
Theor. Probability and Math. Statist. 99 (2019), 91-99
MSC (2010):
Primary 60G70, 62J05
DOI:
https://doi.org/10.1090/tpms/1082
Published electronically:
February 27, 2020
MathSciNet review:
3908658
Full-text PDF
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Abstract: We prove a stronger version of the weak consistency of a minimax estimator of a vector regression parameter and prove a limit theorem for absolute values of extreme residuals constructed from this estimator in a linear regression model and for the uniform design of the regression experiment.
References
- Aleksander Ivanov, Ivan Matsak, and Sergiy Polotskiy, Extreme residuals in regression model. Minimax approach, Mod. Stoch. Theory Appl. 2 (2015), no. 3, 297–308. MR 3407508, DOI https://doi.org/10.15559/15-VMSTA40CNF
- William Feller, An introduction to probability theory and its applications. Vol. II., 2nd ed., John Wiley & Sons, Inc., New York-London-Sydney, 1971. MR 0270403
- B. Gnedenko, Sur la distribution limite du terme maximum d’une série aléatoire, Ann. of Math. (2) 44 (1943), 423–453 (French). MR 8655, DOI https://doi.org/10.2307/1968974
- Janos Galambos, The asymptotic theory of extreme order statistics, John Wiley & Sons, New York-Chichester-Brisbane, 1978. Wiley Series in Probability and Mathematical Statistics. MR 489334
- M. R. Leadbetter, Georg Lindgren, and Holger Rootzén, Extremes and related properties of random sequences and processes, Springer Series in Statistics, Springer-Verlag, New York-Berlin, 1983. MR 691492
- Vladimir M. Zolotarev, Modern theory of summation of random variables, Modern Probability and Statistics, VSP, Utrecht, 1997. MR 1640024
- J. Pfanzagl, On the Measurability and Consistency of Minimum Contrast Estimates, Metrika (1969), no. 14, 249–272.
- O. V. Īvanov and Ī. K. Matsak, Limit theorems for extremal residuals in linear and nonlinear regression models, Teor. Ĭmovīr. Mat. Stat. 86 (2011), 69–80 (Ukrainian, with English, Russian and Ukrainian summaries); English transl., Theory Probab. Math. Statist. 86 (2013), 79–91. MR 2986451, DOI https://doi.org/10.1090/S0094-9000-2013-00890-4
- O. V. Īvanov and Ī. K. Matsak, Limit theorems for extreme residuals in a regression model with observation errors having heavy tails, Teor. Ĭmovīr. Mat. Stat. 88 (2013), 90–97 (Ukrainian, with English, Russian and Ukrainian summaries); English transl., Theory Probab. Math. Statist. 88 (2014), 99–108. MR 3112637, DOI https://doi.org/10.1090/S0094-9000-2014-00921-7
References
- A. V. Ivanov, I. K. Matsak, and S. V. Polotskiy, Extreme residuals in regression model. Minimax approach, Modern stochastics: theory and applications 2 (2015), no. 3, 297–308. MR 3407508
- W. Feller, An Introduction to Probability Theory and its Applications. vol. 2, Wiley, New York–London–Sydney, 1971. MR 0270403
- B. V. Gnedenko, Sur la distribution limit du terme maximum d‘une serie aleatoire, Ann. Math. 44 (1943), 423–453. MR 8655
- J. Galambos, The Asymptotic Theory of Extreme Order Statistics, Wiley, New York, 1978. MR 489334
- M. R. Leadbetter, G. Lindgren, and H. Rootzen, Extremes and Related Properties of Random Sequences and Processes, Springer, New York–Berlin, 1983. MR 691492
- V. M. Zolotarev, Modern Theory of Summation of Random Variable, VSP, Utrecht, 1997. MR 1640024
- J. Pfanzagl, On the Measurability and Consistency of Minimum Contrast Estimates, Metrika (1969), no. 14, 249–272.
- O. V. Ivanov and I. K. Matsak, Limit theorems for extreme residuals in linear and nonlinear regression models, Teor. Imovirnost. ta Matem. Statyst. 86 (2012), 69–80; English transl. in Theor. Probability and Math. Statist. 86 (2013), 79–91. MR 2986451
- O. V. Ivanov and I. K. Matsak, Limit theorems for extreme residuals in regression models with heavy tails of observation errors, Teor. Imovirnost. ta Matem. Statyst. 88 (2013), 85–98; English transl. in Theor. Probability and Math. Statist. 88 (2014), 99–108. MR 3112637
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Additional Information
A. V. Ivanov
Affiliation:
Department of Mathematical Analysis and Probability Theory, Department of Physics and Mathematics, National Technical University of Ukraine “Igor Sikorsky Kyiv Polytechnic Institute”, Peremogy Avenue, 37, Kyiv, 03057 Ukraine
Email:
alexntuu@gmail.com
I. K. Matsak
Affiliation:
Taras Shevchenko National University, Academician Glushkov Avenue, 2, Building 6, Kyiv, 03127 Ukraine
Email:
mik@unicyb.kiev.ua
Keywords:
Linear regression model,
extreme values,
scheme of series,
minimax estimators,
symmetric errors of observations
Received by editor(s):
July 2, 2018
Published electronically:
February 27, 2020
Article copyright:
© Copyright 2020
American Mathematical Society
