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Change Point Estimation by Local Linear Smoothing

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  • Grégoire, Gérard
  • Hamrouni, Zouhir

Abstract

We consider the problem of estimating jump points in smooth curves. Observations (Xi,Yi) i=1,...,n from a random design regression function are given. We focus essentially on the basic situation where a unique change point is present in the regression function. Based on local linear regression, a jump estimate process t-->[gamma](t) is constructed. Our main result is the convergence to a compound Poisson process with drift, of a local dilated-rescaled version of [gamma](t), under a positivity condition regarding the asymmetric kernel involved. This result enables us to prove that our estimate of the jump location converges with exact rate n-1 without any particular assumption regarding the bandwidth hn. Other consequences such as asymptotic normality are investigated and some proposals are provided for an extension of this work to more general situations. Finally we present Monte-Carlo simulations which give evidence for good numerical performance of our procedure.

Suggested Citation

  • Grégoire, Gérard & Hamrouni, Zouhir, 2002. "Change Point Estimation by Local Linear Smoothing," Journal of Multivariate Analysis, Elsevier, vol. 83(1), pages 56-83, October.
    • Handle: RePEc:eee:jmvana:v:83:y:2002:i:1:p:56-83
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    References listed on IDEAS

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    1. 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.
    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. Ward Whitt, 1980. "Some Useful Functions for Functional Limit Theorems," Mathematics of Operations Research, INFORMS, vol. 5(1), pages 67-85, February.
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    Cited by:

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    2. Einmahl, J.H.J. & Gantner, M., 2009. "The Half-Half Plot," Discussion Paper 2009-77, Tilburg University, Center for Economic Research.
    3. Gantner, M., 2010. "Some nonparametric diagnostic statistical procedures and their asymptotic behavior," Other publications TiSEM eb04bdba-bf8a-4f6c-8dd8-9, Tilburg University, School of Economics and Management.
    4. Daniel J. Henderson & Christopher F. Parmeter & Liangjun Su, 2017. "M-Estimation of a Nonparametric Threshold Regression Model," Working Papers 2017-15, University of Miami, Department of Economics.
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    8. Lin, Zhengyan & Li, Degui & Chen, Jia, 2008. "Change point estimators by local polynomial fits under a dependence assumption," Journal of Multivariate Analysis, Elsevier, vol. 99(10), pages 2339-2355, November.
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    11. Porter, Jack & Yu, Ping, 2015. "Regression discontinuity designs with unknown discontinuity points: Testing and estimation," Journal of Econometrics, Elsevier, vol. 189(1), pages 132-147.

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