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Two-stage change-point estimators in smooth regression models

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  • Müller, Hans-Georg
  • Song, Kai-Sheng

Abstract

We consider a fixed design regression model where the regression function is assumed to be smooth, i.e., Lipschitz continuous, except for a point where it has only one-sided limits and a local discontinuity occurs. We propose a two-step estimator for the location of this change point and study its asymptotic convergence properties. In a first step, initial pilot estimates of the change point and associated asymptotically shrinking intervals which contain the true change point with probability converging to 1 are obtained. In the second step, a weighted mean difference depending on the assumed location of the change point is maximized within these intervals and the maximizing argument is then the final change point estimator. It is shown that this estimator attains the rate Op(n-1) in the fixed jump case. In the contiguous case, the estimator attains the rate Op(n-1 [Delta]n-2), where [Delta]n is the sequence of jump sizes which in this case is assumed to converge to 0. For the contiguous case an invariance principle is established. A sequence of appropriately scaled deviation processes is shown to converge to a two-sided Brownian motion with triangular drift.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:stapro:v:34:y:1997:i:4:p:323-335
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    Citations

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    Cited by:

    1. Grégoire, Gérard & Hamrouni, Zouhir, 2002. "Change Point Estimation by Local Linear Smoothing," Journal of Multivariate Analysis, Elsevier, vol. 83(1), pages 56-83, October.
    2. Yujiao Yang & Qiongxia Song, 2014. "Jump detection in time series nonparametric regression models: a polynomial spline approach," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 66(2), pages 325-344, April.
    3. Huh, Jib, 2010. "Detection of a change point based on local-likelihood," Journal of Multivariate Analysis, Elsevier, vol. 101(7), pages 1681-1700, August.
    4. Zhanfeng Wang & Wenxin Liu & Yuanyuan Lin, 2015. "A change-point problem in relative error-based regression," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 24(4), pages 835-856, December.
    5. Irène Gijbels & Alexandre Lambert & Peihua Qiu, 2007. "Jump-Preserving Regression and Smoothing using Local Linear Fitting: A Compromise," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 59(2), pages 235-272, June.
    6. 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.
    7. Shujie Ma & Lijian Yang, 2011. "A jump-detecting procedure based on spline estimation," Journal of Nonparametric Statistics, Taylor & Francis Journals, vol. 23(1), pages 67-81.
    8. Atul Mallik & Moulinath Banerjee & George Michailidis, 2020. "M-estimation in Multistage Sampling Procedures," Sankhya A: The Indian Journal of Statistics, Springer;Indian Statistical Institute, vol. 82(2), pages 261-309, August.
    9. Ferger Dietmar & Klotsche Jens, 2009. "Estimation of split-points in binary regression," Statistics & Risk Modeling, De Gruyter, vol. 27(02), pages 93-128, December.
    10. Jens Klotsche & Andrew T. Gloster, 2012. "Estimating a Meaningful Point of Change," Journal of Educational and Behavioral Statistics, , vol. 37(5), pages 579-600, October.
    11. Huh, J. & Carrière, K. C., 2002. "Estimation of regression functions with a discontinuity in a derivative with local polynomial fits," Statistics & Probability Letters, Elsevier, vol. 56(3), pages 329-343, February.
    12. Kang, Yicheng & Shi, Yueyong & Jiao, Yuling & Li, Wendong & Xiang, Dongdong, 2021. "Fitting jump additive models," Computational Statistics & Data Analysis, Elsevier, vol. 162(C).
    13. Irene Gijbels & Peter Hall & Aloïs Kneip, 1999. "On the Estimation of Jump Points in Smooth Curves," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 51(2), pages 231-251, June.
    14. Yu, Ping & Phillips, Peter C.B., 2018. "Threshold regression with endogeneity," Journal of Econometrics, Elsevier, vol. 203(1), pages 50-68.
    15. 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.
    16. Cui, Yan & Yang, Jun & Zhou, Zhou, 2023. "State-domain change point detection for nonlinear time series regression," Journal of Econometrics, Elsevier, vol. 234(1), pages 3-27.

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