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Smooth change point estimation in regression models with random design

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  • Maik Döring
  • Uwe Jensen

Abstract

We consider the problem of estimating the location of a change point $$\theta _0$$ θ 0 in a regression model. Most change point models studied so far were based on regression functions with a jump. However, we focus on regression functions, which are continuous at $$\theta _0$$ θ 0 . The degree of smoothness $$q_0$$ q 0 has to be estimated as well. We investigate the consistency with increasing sample size $$n$$ n of the least squares estimates $$(\hat{\theta }_n,\hat{q}_n)$$ ( θ ^ n , q ^ n ) of $$(\theta _0, q_0)$$ ( θ 0 , q 0 ) . It turns out that the rates of convergence of $$\hat{\theta }_n$$ θ ^ n depend on $$q_0$$ q 0 : for $$q_0$$ q 0 greater than $$1/2$$ 1 / 2 we have a rate of $$\sqrt{n}$$ n and the asymptotic normality property; for $$q_0$$ q 0 less than $$1/2$$ 1 / 2 the rate is $$\displaystyle n^{1/(2q_0+1)}$$ n 1 / ( 2 q 0 + 1 ) and the change point estimator converges to a maximizer of a Gaussian process; for $$q_0$$ q 0 equal to $$1/2$$ 1 / 2 the rate is $$\sqrt{n \cdot \mathrm{ln}(n)}$$ n · ln ( n ) . Interestingly, in the last case the limiting distribution is also normal. Copyright The Institute of Statistical Mathematics, Tokyo 2015

Suggested Citation

  • Maik Döring & Uwe Jensen, 2015. "Smooth change point estimation in regression models with random design," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 67(3), pages 595-619, June.
  • Handle: RePEc:spr:aistmt:v:67:y:2015:i:3:p:595-619
    DOI: 10.1007/s10463-014-0467-8
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    References listed on IDEAS

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    1. Koul, Hira L. & Qian, Lianfen & Surgailis, Donatas, 2003. "Asymptotics of M-estimators in two-phase linear regression models," Stochastic Processes and their Applications, Elsevier, vol. 103(1), pages 123-154, January.
    2. Müller, Hans-Georg & Song, Kai-Sheng, 1997. "Two-stage change-point estimators in smooth regression models," Statistics & Probability Letters, Elsevier, vol. 34(4), pages 323-335, June.
    3. Jushan Bai, 1997. "Estimation Of A Change Point In Multiple Regression Models," The Review of Economics and Statistics, MIT Press, vol. 79(4), pages 551-563, November.
    4. Aue, Alexander & Steinebach, Josef, 2002. "A note on estimating the change-point of a gradually changing stochastic process," Statistics & Probability Letters, Elsevier, vol. 56(2), pages 177-191, January.
    5. Astrid Dempfle & Winfried Stute, 2002. "Nonparametric estimation of a discontinuity in regression," Statistica Neerlandica, Netherlands Society for Statistics and Operations Research, vol. 56(2), pages 233-242, May.
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    Cited by:

    1. Zhao, Wenbiao & Zhu, Lixing, 2024. "Detecting change structures of nonparametric regressions," Computational Statistics & Data Analysis, Elsevier, vol. 190(C).
    2. S. Dachian & N. Kordzakhia & Yu. A. Kutoyants & A. Novikov, 2018. "Estimation of cusp location of stochastic processes: a survey," Statistical Inference for Stochastic Processes, Springer, vol. 21(2), pages 345-362, July.
    3. O. V. Chernoyarov & S. Dachian & Yu. A. Kutoyants, 2018. "On parameter estimation for cusp-type signals," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 70(1), pages 39-62, February.
    4. Hyunju Son & Youyi Fong, 2021. "Fast grid search and bootstrap‐based inference for continuous two‐phase polynomial regression models," Environmetrics, John Wiley & Sons, Ltd., vol. 32(3), May.
    5. Maria Mohr & Leonie Selk, 2020. "Estimating change points in nonparametric time series regression models," Statistical Papers, Springer, vol. 61(4), pages 1437-1463, August.
    6. Marie Hušková & Zuzana Prášková & Josef G. Steinebach, 2022. "Estimating a gradual parameter change in an AR(1)-process," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 85(7), pages 771-808, October.

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