Large deviations of regression parameter estimate in the models with stationary sub-Gaussian noise

Author:
A. V. Ivanov

Original publication:
Teoriya Imovirnostei ta Matematichna Statistika, tom **95** (2016).

Journal:
Theor. Probability and Math. Statist. **95** (2017), 99-108

MSC (2010):
Primary 60G50, 65B10, 60G15; Secondary 40A05

DOI:
https://doi.org/10.1090/tpms/1024

Published electronically:
February 28, 2018

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Exponential bounds for probabilities of large deviations of nonlinear regression parameter least squares estimate in the models with jointly strictly sub-Gaussian random noise are obtained.

- [1]
A.
V. Ivanov,
*Asymptotic expansions for distributions of least squares estimators of parameters of nonlinear regression*, Teor. Verojatnost. i Primenen.**21**(1976), no. 3, 571–583 (Russian, with English summary). MR**0428547** - [2]
B.
L. S. Prakasa Rao,
*On the exponential rate of convergence of the least squares estimator in the nonlinear regression model with Gaussian errors*, Statist. Probab. Lett.**2**(1984), no. 3, 139–142. MR**747613**, https://doi.org/10.1016/0167-7152(84)90004-X - [3]
Arthur
Sieders and Kacha
Dzhaparidze,
*A large deviation result for parameter estimators and its application to nonlinear regression analysis*, Ann. Statist.**15**(1987), no. 3, 1031–1049. MR**902244**, https://doi.org/10.1214/aos/1176350491 - [4]
I.
A. Ibragimov and R.
Z. Has′minskiĭ,
*Statistical estimation*, Applications of Mathematics, vol. 16, Springer-Verlag, New York-Berlin, 1981. Asymptotic theory; Translated from the Russian by Samuel Kotz. MR**620321** - [5]
Alexander
V. Ivanov,
*Asymptotic theory of nonlinear regression*, Mathematics and its Applications, vol. 389, Kluwer Academic Publishers Group, Dordrecht, 1997. MR**1472234** - [6]
A.
V. Ivanov and N.
N. Leonenko,
*Statistical analysis of random fields*, Mathematics and its Applications (Soviet Series), vol. 28, Kluwer Academic Publishers Group, Dordrecht, 1989. With a preface by A. V. Skorokhod; Translated from the Russian by A. I. Kochubinskiĭ. MR**1009786** - [7]
B.
L. S. Prakasa Rao,
*The rate of convergence of the least squares estimator in a nonlinear regression model with dependent errors*, J. Multivariate Anal.**14**(1984), no. 3, 315–322. MR**747260**, https://doi.org/10.1016/0047-259X(84)90036-8 - [8]
Shu
He Hu,
*A large deviation result for the least squares estimators in nonlinear regression*, Stochastic Process. Appl.**47**(1993), no. 2, 345–352. MR**1239845**, https://doi.org/10.1016/0304-4149(93)90022-V - [9]
Wenzhi
Yang and Shuhe
Hu,
*Large deviation for a least squares estimator in a nonlinear regression model*, Statist. Probab. Lett.**91**(2014), 135–144. MR**3208127**, https://doi.org/10.1016/j.spl.2014.04.022 - [10]
Xufeng
Huang, Xufei
Tang, Xin
Deng, and Xuejun
Wang,
*The large deviation for the least squares estimator of nonlinear regression model based on WOD errors*, J. Inequal. Appl. , posted on (2016), Paper No. 125, 11. MR**3489878**, https://doi.org/10.1186/s13660-016-1064-6 - [11]
J. Pfanzagl,
*On the measurability and consistency of minimum contrast estimates*, J. Metrika**14**(1969), 249-272. - [12]
V.
V. Buldygin and Yu.
V. Kozachenko,
*Metric characterization of random variables and random processes*, Translations of Mathematical Monographs, vol. 188, American Mathematical Society, Providence, RI, 2000. Translated from the 1998 Russian original by V. Zaiats. MR**1743716** - [13]
William
Feller,
*An introduction to probability theory and its applications. Vol. I*, John Wiley and Sons, Inc., New York; Chapman and Hall, Ltd., London, 1957. 2nd ed. MR**0088081** - [14]
I.
I. Gikhman and A.
V. Skorokhod,
*Introduction to the theory of random processes*, Dover Publications, Inc., Mineola, NY, 1996. Translated from the 1965 Russian original; Reprint of the 1969 English translation; With a preface by Warren M. Hirsch. MR**1435501** - [15]
Peter
J. Brockwell and Richard
A. Davis,
*Introduction to time series and forecasting*, 2nd ed., Springer Texts in Statistics, Springer-Verlag, New York, 2002. With 1 CD-ROM (Windows). MR**1894099**

Retrieve articles in *Theory of Probability and Mathematical Statistics*
with MSC (2010):
60G50,
65B10,
60G15,
40A05

Retrieve articles in all journals with MSC (2010): 60G50, 65B10, 60G15, 40A05

Additional Information

**A. V. Ivanov**

Affiliation:
Department of Mathematical Analysis and Probability Theory, Faculty of Physics and Mathematics, NTUU“KPI”, Kyiv, Ukraine

Email:
alexntuu@gmail.com

DOI:
https://doi.org/10.1090/tpms/1024

Keywords:
Large deviations,
least squares estimate,
nonlinear regression,
discrete white sub-Gaussian noise,
spectral density

Received by editor(s):
August 29, 2016

Published electronically:
February 28, 2018

Article copyright:
© Copyright 2018
American Mathematical Society