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On smooth change-point location estimation for Poisson Processes

Author

Listed:
  • Arij Amiri

    (Univ. Lille, CNRS, UMR 8524 — Laboratoire Paul Painlevé)

  • Sergueï Dachian

    (Univ. Lille, CNRS, UMR 8524 — Laboratoire Paul Painlevé
    Tomsk State University)

Abstract

We are interested in estimating the location of what we call “smooth change-point” from n independent observations of an inhomogeneous Poisson process. The smooth change-point is a transition of the intensity function of the process from one level to another which happens smoothly, but over such a small interval, that its length $$\delta _n$$ δ n is considered to be decreasing to 0 as $$n\rightarrow +\infty $$ n → + ∞ . We show that if $$\delta _n$$ δ n goes to zero slower than 1/n, our model is locally asymptotically normal (with a rather unusual rate $$\sqrt{\delta _n/n}$$ δ n / n ), and the maximum likelihood and Bayesian estimators are consistent, asymptotically normal and asymptotically efficient. If, on the contrary, $$\delta _n$$ δ n goes to zero faster than 1/n, our model is non-regular and behaves like a change-point model. More precisely, in this case we show that the Bayesian estimators are consistent, converge at rate 1/n, have non-Gaussian limit distributions and are asymptotically efficient. All these results are obtained using the likelihood ratio analysis method of Ibragimov and Khasminskii, which equally yields the convergence of polynomial moments of the considered estimators. However, in order to study the maximum likelihood estimator in the case where $$\delta _n$$ δ n goes to zero faster than 1/n, this method cannot be applied using the usual topologies of convergence in functional spaces. So, this study should go through the use of an alternative topology and will be considered in a future work.

Suggested Citation

  • Arij Amiri & Sergueï Dachian, 2021. "On smooth change-point location estimation for Poisson Processes," Statistical Inference for Stochastic Processes, Springer, vol. 24(3), pages 499-524, October.
  • Handle: RePEc:spr:sistpr:v:24:y:2021:i:3:d:10.1007_s11203-021-09240-w
    DOI: 10.1007/s11203-021-09240-w
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    References listed on IDEAS

    as
    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.
    2. Ji Hwan Cha & Maxim Finkelstein, 2018. "Point Processes for Reliability Analysis," Springer Series in Reliability Engineering, Springer, number 978-3-319-73540-5, March.
    3. O. V. Chernoyarov & Yu. A. Kutoyants, 2020. "Poisson source localization on the plane: the smooth case," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 83(4), pages 411-435, May.
    4. S. Dachian & Yu. A. Kutoyants & L. Yang, 2016. "On hypothesis testing for Poisson processes. Singular cases," Communications in Statistics - Theory and Methods, Taylor & Francis Journals, vol. 45(23), pages 6833-6859, December.
    5. S. Dachian, 2003. "Estimation of Cusp Location by Poisson Observations," Statistical Inference for Stochastic Processes, Springer, vol. 6(1), pages 1-14, January.
    6. S. Dachian & Yu. A. Kutoyants & L. Yang, 2016. "On hypothesis testing for Poisson processes: Regular case," Communications in Statistics - Theory and Methods, Taylor & Francis Journals, vol. 45(23), pages 6816-6832, December.
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