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Inference for nonstationary time series of counts with application to change-point problems

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Abstract

We consider an integer-valued time series \((Y_t)_{t\in {\mathbb {Z}}}\) where the model after a time \(k^*\) is Poisson autoregressive with the conditional mean that depends on a parameter \(\theta ^*\in \varTheta \subset {\mathbb {R}}^d\). The structure of the process before \(k^*\) is unknown; it could be any other integer-valued process, that is, \((Y_t)_{t\in {\mathbb {Z}}}\) could be nonstationary. It is established that the maximum likelihood estimator of \(\theta ^*\) computed on the nonstationary observations is consistent and asymptotically normal. Subsequently, we carry out the sequential change-point detection in a large class of Poisson autoregressive models, and propose a monitoring scheme for detecting change. The procedure is based on an updated estimator, which is computed without the historical observations. The above results of inference in a nonstationary setting are applied to prove the consistency of the proposed procedure. A simulation study as well as a real data application are provided.

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Acknowledgements

The authors are grateful to the three anonymous Referees for many relevant suggestions and comments which helped to improve the contents of this paper.

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Correspondence to William Kengne.

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William Kengne was developed within the ANR BREAKRISK: ANR-17-CE26-0001-01 and the CY Initiative of Excellence (Grant “Investissements d’Avenir” ANR-16-IDEX-0008), Project “EcoDep” PSI-AAP2020-0000000013.

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Kengne, W., Ngongo, I.S. Inference for nonstationary time series of counts with application to change-point problems. Ann Inst Stat Math 74, 801–835 (2022). https://doi.org/10.1007/s10463-021-00815-1

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  • DOI: https://doi.org/10.1007/s10463-021-00815-1

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