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On a Poissonian change-point model with variable jump size

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  • Sergueï Dachian
  • Lin Yang

Abstract

A model of Poissonian observations having a jump (change-point) in the intensity function is considered. Two cases are studied. The first one corresponds to the situation when the jump size converges to a non-zero limit, while in the second one the limit is zero. The limiting likelihood ratios in these two cases are quite different. In the first case, like in the case of a fixed jump size, the normalized likelihood ratio converges to a log Poisson process. In the second case, the normalized likelihood ratio converges to a log Wiener process, and so, the statistical problems of parameter estimation and hypothesis testing are asymptotically equivalent in this case to the well known problems of change-point estimation and testing for the model of a signal in white Gaussian noise. The properties of the maximum likelihood and Bayesian estimators, as well as those of the general likelihood ratio, Wald’s and Bayesian tests are deduced from the convergence of normalized likelihood ratios. The convergence of the moments of the estimators is also established. The obtained theoretical results are illustrated by numerical simulations. Copyright Springer Science+Business Media Dordrecht 2015

Suggested Citation

  • Sergueï Dachian & Lin Yang, 2015. "On a Poissonian change-point model with variable jump size," Statistical Inference for Stochastic Processes, Springer, vol. 18(2), pages 127-150, July.
    • Handle: RePEc:spr:sistpr:v:18:y:2015:i:2:p:127-150
    DOI: 10.1007/s11203-014-9109-2
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    References listed on IDEAS

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    1. 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.
    2. S. Dachian, 2003. "Estimation of Cusp Location by Poisson Observations," Statistical Inference for Stochastic Processes, Springer, vol. 6(1), pages 1-14, January.
    3. Sergueï Dachian & Ilia Negri, 2011. "On compound Poisson processes arising in change-point type statistical models as limiting likelihood ratios," Statistical Inference for Stochastic Processes, Springer, vol. 14(3), pages 255-271, October.
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