Asymptotic properties of 𝑀-estimators of parameters of a nonlinear regression model with a random noise whose spectrum is singular

Ivanov, A.; Orlovskyi, I.

doi:10.1090/tpms/993

Asymptotic properties of -estimators of parameters of a nonlinear regression model with a random noise whose spectrum is singular

Authors: A. V. Ivanov and I. V. Orlovskyi
Translated by: S. Kvasko
Original publication: Teoriya Imovirnostei ta Matematichna Statistika, tom 93 (2015).
Journal: Theor. Probability and Math. Statist. 93 (2016), 33-49
MSC (2010): Primary 62J02; Secondary 62J99
DOI: https://doi.org/10.1090/tpms/993
Published electronically: February 7, 2017
MathSciNet review: 3553438
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Abstract: Time continuous nonlinear regression model with a noise being a nonlinearly transformed Gaussian stationary process with a singular spectrum is considered in the paper. Sufficient conditions for the asymptotic normality of the -estimator are found for the vector parameter in this model.

1. Peter J. Huber, Robust regression: asymptotics, conjectures and Monte Carlo, Ann. Statist. 1 (1973), 799–821. MR 0356373
2. Peter J. Huber, Robust statistics, John Wiley & Sons, Inc., New York, 1981. Wiley Series in Probability and Mathematical Statistics. MR 606374
3. Frank R. Hampel, Elvezio M. Ronchetti, Peter J. Rousseeuw, and Werner A. Stahel, Robust statistics, Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics, John Wiley & Sons, Inc., New York, 1986. The approach based on influence functions. MR 829458
4. Hira L. Koul, 𝑀-estimators in linear models with long range dependent errors, Statist. Probab. Lett. 14 (1992), no. 2, 153–164. MR 1173413, https://doi.org/10.1016/0167-7152(92)90079-K
5. Hira L. Koul, Asymptotics of 𝑀-estimators in non-linear regression with long-range dependent errors, Athens Conference on Applied Probability and Time Series Analysis, Vol. II (1995), Lect. Notes Stat., vol. 115, Springer, New York, 1996, pp. 272–290. MR 1466752, https://doi.org/10.1007/978-1-4612-2412-9_20
6. Hira L. Koul and Kanchan Mukherjee, Regression quantiles and related processes under long range dependent errors, J. Multivariate Anal. 51 (1994), no. 2, 318–337. MR 1321301, https://doi.org/10.1006/jmva.1994.1065
7. Liudas Giraitis, Hira L. Koul, and Donatas Surgailis, Asymptotic normality of regression estimators with long memory errors, Statist. Probab. Lett. 29 (1996), no. 4, 317–335. MR 1409327, https://doi.org/10.1016/0167-7152(95)00188-3
8. Hira L. Koul and Donatas Surgailis, Asymptotic expansion of 𝑀-estimators with long-memory errors, Ann. Statist. 25 (1997), no. 2, 818–850. MR 1439325, https://doi.org/10.1214/aos/1031833675
9. Hira L. Koul and Donatas Surgailis, Second-order behavior of 𝑀-estimators in linear regression with long-memory errors, J. Statist. Plann. Inference 91 (2000), no. 2, 399–412. Prague Workshop on Perspectives in Modern Statistical Inference: Parametrics, Semi-parametrics, Non-parametrics (1998). MR 1814792, https://doi.org/10.1016/S0378-3758(00)00190-7
10. H. L. Koul and D. Surgailis, Robust estimators in regression models with long memory errors, Theory and applications of long-range dependence, Birkhäuser Boston, Boston, MA, 2003, pp. 339–353. MR 1957498
11. Liudas Giraitis, Hira L. Koul, and Donatas Surgailis, Asymptotic normality of regression estimators with long memory errors, Statist. Probab. Lett. 29 (1996), no. 4, 317–335. MR 1409327, https://doi.org/10.1016/0167-7152(95)00188-3
12. Hira L. Koul, Richard T. Baillie, and Donatas Surgailis, Regression model fitting with a long memory covariate process, Econometric Theory 20 (2004), no. 3, 485–512. MR 2061725, https://doi.org/10.1017/S0266466604203036
13. A. V. Ivanov and N. N. Leonenko, Asymptotic behavior of 𝑀-estimators in continuous-time non-linear regression with long-range dependent errors, Random Oper. Stochastic Equations 10 (2002), no. 3, 201–222. MR 1923424, https://doi.org/10.1515/rose.2002.10.3.201
14. A. Ivanov and N. Leonenko, Robust estimators in non-linear regression models with long-range dependence, Optimal design and related areas in optimization and statistics, Springer Optim. Appl., vol. 28, Springer, New York, 2009, pp. 193–221. MR 2513352, https://doi.org/10.1007/978-0-387-79936-0_9
15. Alexander V. Ivanov, Asymptotic properties of 𝐿_{𝑝}-estimators, Theory Stoch. Process. 14 (2008), no. 1, 60–68. MR 2479706
16. A. V. Ivanov and I. V. Orlovsky, L-estimates in nonlinear regression with long-range dependence, Theory Stoch. Process. 7(23) (2002), no. 3-4, 38-49.
17. A. V. Ivanov and I. V. Orlovskyi, Consistency of -estimators in nonlinear regression models with continuous time, Naukovi visti NTUU ``KPI'' 4(42) (2005), 140-147. (Ukrainian)
18. A. V. Ivanov and I. V. Orlovskyi, The uniqueness of -estimators of parameters in nonlinear regression models, Naukovi Visti NTUU ``KPI'' 4(66) (2009), 135-141. (Ukrainian)
19. Ī. M. Savich, Consistency of quantile estimators in regression models with long-range dependent noise, Teor. Ĭmovīr. Mat. Stat. 82 (2010), 128–136 (Ukrainian, with English, Russian and Ukrainian summaries); English transl., Theory Probab. Math. Statist. 82 (2011), 129–138. MR 2790488, https://doi.org/10.1090/S0094-9000-2011-00832-0
20. Igor V. Orlovsky, 𝑀-estimates in nonlinear regression with weak dependence, Theory Stoch. Process. 9 (2003), no. 1-2, 108–122. MR 2080017
21. Alexander Ivanov and Igor V. Orlovsky, Consistency of 𝑀-estimates in general nonlinear regression models, Theory Stoch. Process. 13 (2007), no. 1-2, 86–97. MR 2343813
22. I. V. Orlovsky, Asymptotic properties of -estimators of parameters in nonlinear regression models, Abstract of candidate thesis (Physics and Mathematics), Kyiv National Taras Shevchenko University, Kyiv, 2007. (Ukrainian)
23. Alexander V. Ivanov, Nikolai Leonenko, María D. Ruiz-Medina, and Irina N. Savich, Limit theorems for weighted nonlinear transformations of Gaussian stationary processes with singular spectra, Ann. Probab. 41 (2013), no. 2, 1088–1114. MR 3077537, https://doi.org/10.1214/12-AOP775
24. Yu. V. Goncharenko and S. I. Lyashko, Brouwer's Theorem, ``KIĬ'', Kiev, 2000. (Russian)
25. V. V. Anh, V. P. Knopova, and N. N. Leonenko, Continuous-time stochastic processes with cyclical long-range dependence, Aust. N. Z. J. Stat. 46 (2004), no. 2, 275–296. MR 2076396, https://doi.org/10.1111/j.1467-842X.2004.00329.x
26. A. V. Ivanov, N. N. Leonenko, M. D. Ruiz-Medina, and B. M. Zhurakovsky, Estimation of harmonic component in regression with cyclically dependent errors, Statistics 49 (2015), no. 1, 156–186. MR 3304373, https://doi.org/10.1080/02331888.2013.864656
27. Ulf Grenander, On the estimation of regression coefficients in the case of an autocorrelated disturbance, Ann. Math. Statistics 25 (1954), 252–272. MR 62402, https://doi.org/10.1214/aoms/1177728784
28. Il′dar Abdullovich Ibragimov and Y. A. Rozanov, Gaussian random processes, Applications of Mathematics, vol. 9, Springer-Verlag, New York-Berlin, 1978. Translated from the Russian by A. B. Aries. MR 543837
29. Patrick Billingsley, Convergence of probability measures, John Wiley & Sons, Inc., New York-London-Sydney, 1968. MR 0233396
30. A. V. Ivanov and I. M. Savych, The $\mu$ -admissibility of a spectral density for long range dependent random noise in nonlinear regression models, Naukovi visti NTUU ``KPI'' 1 (2009), 143-148. (Ukrainian)
31. N. N. Leonenko and A. V. Ivanov, Statistical Analysis of Random Fields, ``Vyshcha Shkola'', Kiev, 1986; English transl., Kluwer Academic Publishers Group, Dordrecht, 1989.
32. R. N. Bhattacharya and R. Ranga Rao, Normal Approximation and Asymptotic Expansions, John Wiley & Sons, New York-London-Sydney, 1976.
33. J. H. Wilkinson, The Algebraic Eigen Value Problem, Clarendon Press, Oxford, 1962.