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Asymptotics of M-estimators in two-phase linear regression models

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  • Koul, Hira L.
  • Qian, Lianfen
  • Surgailis, Donatas

Abstract

This paper discusses the consistency and limiting distributions of a class of M-estimators in two-phase random design linear regression models where the regression function is discontinuous at the change-point with a fixed jump size. The consistency rate of an M-estimator for the change-point parameter r is shown to be n while it is n1/2 for the coefficient parameter estimators, where n denotes the sample size. The normalized M-process is shown to be uniformly locally asymptotically equivalent to the sum of a quadratic form in the coefficient parameter vector and a jump point process in the change-point parameter, in probability. These results are then used to obtain the joint weak convergence of the M-estimators. In particular, is shown to converge weakly to a random variable which minimizes a compound Poisson process, a suitably standardized coefficient parameter M-estimator vector is shown to be asymptotically normal, and independent of .

Suggested Citation

  • 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.
    • Handle: RePEc:eee:spapps:v:103:y:2003:i:1:p:123-154
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    References listed on IDEAS

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    1. 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.
    2. Jushan Bai & Pierre Perron, 1998. "Estimating and Testing Linear Models with Multiple Structural Changes," Econometrica, Econometric Society, vol. 66(1), pages 47-78, January.
    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:

    1. Chung-Ming Kuan & Christos Michalopoulos & Zhijie Xiao, 2017. "Quantile Regression on Quantile Ranges – A Threshold Approach," Journal of Time Series Analysis, Wiley Blackwell, vol. 38(1), pages 99-119, January.
    2. Gill, Ryan & Lee, Kiseop & Song, Seongjoo, 2007. "Computation of estimates in segmented regression and a liquidity effect model," Computational Statistics & Data Analysis, Elsevier, vol. 51(12), pages 6459-6475, August.
    3. Dupuy, Jean-François, 2006. "Estimation in a change-point hazard regression model," Statistics & Probability Letters, Elsevier, vol. 76(2), pages 182-190, January.
    4. Fujii, Takayuki, 2008. "On weak convergence of the likelihood ratio process in multi-phase regression models," Statistics & Probability Letters, Elsevier, vol. 78(14), pages 2066-2074, October.
    5. Yury Kutoyants, 2012. "On identification of the threshold diffusion processes," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 64(2), pages 383-413, April.
    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. Ferger Dietmar & Klotsche Jens, 2009. "Estimation of split-points in binary regression," Statistics & Risk Modeling, De Gruyter, vol. 27(2), pages 93-128, December.
    8. Fuxia Cheng & Hira L. Koul, 2023. "An analog of Bickel–Rosenblatt test for fitting an error density in the two phase linear regression model," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 86(1), pages 27-56, January.
    9. Gabriela Ciuperca, 2011. "Penalized least absolute deviations estimation for nonlinear model with change-points," Statistical Papers, Springer, vol. 52(2), pages 371-390, May.
    10. Perera, Indeewara & Koul, Hira L., 2017. "Fitting a two phase threshold multiplicative error model," Journal of Econometrics, Elsevier, vol. 197(2), pages 348-367.
    11. Müller, Ursula U. & Schick, Anton & Wefelmeyer, Wolfgang, 2015. "Estimators in step regression models," Statistics & Probability Letters, Elsevier, vol. 100(C), pages 124-129.
    12. Zhang, Li-Xin & Chan, Wai-Sum & Cheung, Siu-Hung & Hung, King-Chi, 2009. "A note on the consistency of a robust estimator for threshold autoregressive processes," Statistics & Probability Letters, Elsevier, vol. 79(6), pages 807-813, March.
    13. Ciuperca, Gabriela, 2009. "The M-estimation in a multi-phase random nonlinear model," Statistics & Probability Letters, Elsevier, vol. 79(5), pages 573-580, March.
    14. Ciuperca, Gabriela, 2011. "A general criterion to determine the number of change-points," Statistics & Probability Letters, Elsevier, vol. 81(8), pages 1267-1275, August.

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