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Estimation of cusp location of stochastic processes: a survey

Author

Listed:
  • S. Dachian

    (University of Lille
    National Research University “MPEI”)

  • N. Kordzakhia

    (Macquarie University)

  • Yu. A. Kutoyants

    (Le Mans University
    National Research University “MPEI”
    Tomsk State University)

  • A. Novikov

    (University of Technology Sydney)

Abstract

We present a review of some recent results on estimation of location parameter for several models of observations with cusp-type singularity at the change point. We suppose that the cusp-type models fit better to the real phenomena described usually by change point models. The list of models includes Gaussian, inhomogeneous Poisson, ergodic diffusion processes, time series and the classical case of i.i.d. observations. We describe the properties of the maximum likelihood and Bayes estimators under some asymptotic assumptions. The asymptotic efficiency of estimators are discussed as well and the results of some numerical simulations are presented. We provide some heuristic arguments which demonstrate the convergence of log-likelihood ratios in the models under consideration to the fractional Brownian motion.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:sistpr:v:21:y:2018:i:2:d:10.1007_s11203-018-9171-2
    DOI: 10.1007/s11203-018-9171-2
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    References listed on IDEAS

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    1. S. Dachian, 2003. "Estimation of Cusp Location by Poisson Observations," Statistical Inference for Stochastic Processes, Springer, vol. 6(1), pages 1-14, January.
    2. 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.
    3. Prakasa Rao, B. L. S., 2004. "Estimation of cusp in nonregular nonlinear regression models," Journal of Multivariate Analysis, Elsevier, vol. 88(2), pages 243-251, February.
    4. Fujii, Takayuki, 2010. "An extension of cusp estimation problem in ergodic diffusion processes," Statistics & Probability Letters, Elsevier, vol. 80(9-10), pages 779-783, May.
    5. Pflug, Georg, 1982. "A statistically important Gaussian Process," Stochastic Processes and their Applications, Elsevier, vol. 13(1), pages 45-57, July.
    6. 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.
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    Cited by:

    1. O. V. Chernoyarov & S. Dachian & Yu. A. Kutoyants, 2020. "Poisson source localization on the plane: cusp case," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 72(5), pages 1137-1157, October.

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