On Bayesian estimators in misspecified change-point problems for Poisson process

Dabye, Ali S.; Farinetto, Christian; Kutoyants, Yury A.

doi:10.1016/s0167-7152(02)00200-6

Cited by 6 publications

(6 citation statements)

References 4 publications

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“…In this discussion, we have only illustrated the robustness of the estimators for the simple toy model, but such a result is typical for many statistical models containing jumps. Note that similar results were also obtained for ergodic diffusion processes and inhomogeneous Poisson processes in Kutoyants (2004), Dabye and Kutoyants (2001) and Dabye et al (2003). It is anticipated for many related problems concerning threshold time series, similar robustness issues can be considered and results in this direction would bear special importance in applications.…”

Section: Misspecificationsupporting

confidence: 74%

Recent Developments of Threshold Estimation for Nonlinear Time Series

Hang

Kutoyants

2010

JJSS

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In this article, several important problems of threshold estimation in a Bayesian framework for nonlinear time series models are discussed. The paper starts with the issue of calculating the maximum likelihood and the Bayesian estimators for threshold autoregressive models. It turns out that the asymptotic efficiency of the Bayesian estimators in this type of singular estimation problems is superior than the maximum likelihood estimators. To illustrate the properties of these estimators and to explain the proposed method, the paper begins with the study of a linear threshold autoregressive model with i.i.d. Gaussian noise. The paper then extends the idea to other nonlinear and non-Gaussian models and illustrates the paradigm of limiting likelihood ratio, which is applicable to a much wider class of nonlinear models. The article also investigates the robustness issue and the possibility of restricting the observation window by narrow bands, which allows one to obtain asymptotically efficient estimators. Finally, the paper indicates how these results can be generalized from a TAR(1) model to a higher-order TAR(p) model with multiple thresholds. The paper concludes with a discussion of other related problems and illustrates the methodology by numerical simulations.

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Section: Misspecificationsupporting

confidence: 74%

Recent Developments of Threshold Estimation for Nonlinear Time Series

Hang

Kutoyants

2010

JJSS

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“…It can be shown that the rate of convergence is essentially better than in regular case, and n θn − ϑ 0 converges in distribution to some random variable (see similar results in Dabye and Kutoyants [3], and Dabye, Farinetto and Kutoyants [2]).…”

Section: Misspecified Modelmentioning

confidence: 54%

“…We suppose that the intensity is bounded positive function and hence the likelihood ratio function for this parametric family is (2) and the MLEθ n and BEθ n for quadratic loss function and prior density p (θ) , θ ∈ Θ (positive, continuous on Θ) are defined by the same equations (1). Regularity Conditions:…”

Section: Regular Casementioning

confidence: 99%

“…Suppose now that the parametric family {λ ϑ , ϑ ∈ Θ} does not correspond to the observed process X n , i.e., the value ϑ 0 ∈ Θ, such that λ * = λ ϑ 0 does not exist, but the statistician nevertheless uses this model to estimate the parameter ϑ (no true model case), i.e., he (or she) calculates the likelihood ratio function by (2), where X n are observations of the Poisson process of intensity function λ * (•). It can be shown that the MLE and BE converge to the value…”

Section: Misspecified Modelmentioning

confidence: 99%

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On properties of estimators in nonregular situations for Poisson processes

Kutoyants

2009

J Math Sci

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We consider the problem of parameter estimation by observations of inhomogeneous Poisson process. It is well-known that if the regularity conditions are fulfilled then the maximum likelihood and bayesian estimators are consistent, asymptotically normal and asymptotically efficient. These regularity conditions can be roughly presented as follows: a) the intensity function of observed process belongs to known parametric family of functions, b) the model is identifiable, c) the Fisher information is positive continuous function, d) the intensity function is sufficiently smooth with respect to the unknown parameter, e) this parameter is an interior point of the interval. We are interested in the properties of estimators when these regularity conditions are not fulfilled. More precisely, we preset a review of the results which correspond to the rejection of these conditions one by one and we show how the properties of the MLE and Bayesian estimators change. The proofs of these results are essentially based on some general results by Ibragimov and Khasminskii. AMS 1991 Classification: 62M05.

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“…Various types of change-point problems in various models are widely considered; for example, see Aue and Steinebach (2002), Pons (2002), Dabye et al (2003), and Xie et al (2004). Statistical inference for change-point is becoming more and more important.…”

Section: Discussionmentioning

confidence: 99%

Parameter Estimation of Some NHPP Software Reliability Models with Change-Point

Wang

2005

Communications in Statistics - Simulation and Computation

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The nonhomogeneous Poisson process (NHPP) model is an important class of software reliability models and is widely used in software reliability engineering. The failure intensity function is usually assumed to be continuous and smooth. However, in many realistic situations, the failure intensity may be not continuous for many possible causes, such as the change in running environment, testing strategy, or resource allocation. The change-point and other parameters are often unknown and to be estimated from the observed failure data. In this article we constructed a method of the type of maximum likelihood estimation, which can be applied in the case that the change-point is not necessarily the observation time point and in the case that the data is grouped. Furthermore, if the failure intensity function is completely unknown, we designed a nonparametric method for estimating the change-point.

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On Bayesian estimators in misspecified change-point problems for Poisson process

Cited by 6 publications

References 4 publications

Recent Developments of Threshold Estimation for Nonlinear Time Series

Recent Developments of Threshold Estimation for Nonlinear Time Series

On properties of estimators in nonregular situations for Poisson processes

Parameter Estimation of Some NHPP Software Reliability Models with Change-Point

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