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Modelling and diagnostic tests for Poisson and negative-binomial count time series

Author

Listed:
  • Boris Aleksandrov

    (Helmut Schmidt University)

  • Christian H. Weiß

    (Helmut Schmidt University)

  • Simon Nik

    (Helmut Schmidt University)

  • Maxime Faymonville

    (TU Dortmund University)

  • Carsten Jentsch

    (TU Dortmund University)

Abstract

When modelling unbounded counts, their marginals are often assumed to follow either Poisson (Poi) or negative binomial (NB) distributions. To test such null hypotheses, we propose goodness-of-fit (GoF) tests based on statistics relying on certain moment properties. By contrast to most approaches proposed in the count-data literature so far, we do not restrict ourselves to specific low-order moments, but consider a flexible class of functions of generalized moments to construct model-diagnostic tests. These cover GoF-tests based on higher-order factorial moments, which are particularly suitable for the Poi- or NB-distribution where simple closed-form expressions for factorial moments of any order exist, but also GoF-tests relying on the respective Stein’s identity for the Poi- or NB-distribution. In the time-dependent case, under mild mixing conditions, we derive the asymptotic theory for GoF tests based on higher-order factorial moments for a wide family of stationary processes having Poi- or NB-marginals, respectively. This family also includes a type of NB-autoregressive model, where we provide clarification of some confusion caused in the literature. Additionally, for the case of independent and identically distributed counts, we prove asymptotic normality results for GoF-tests relying on a Stein identity, and we briefly discuss how its statistic might be used to define an omnibus GoF-test. The performance of the tests is investigated with simulations for both asymptotic and bootstrap implementations, also considering various alternative scenarios for power analyses. A data example of daily counts of downloads of a TeX editor is used to illustrate the application of the proposed GoF-tests.

Suggested Citation

  • Boris Aleksandrov & Christian H. Weiß & Simon Nik & Maxime Faymonville & Carsten Jentsch, 2024. "Modelling and diagnostic tests for Poisson and negative-binomial count time series," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 87(7), pages 843-887, October.
    • Handle: RePEc:spr:metrik:v:87:y:2024:i:7:d:10.1007_s00184-023-00934-0
    DOI: 10.1007/s00184-023-00934-0
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    References listed on IDEAS

    as
    1. Schweer, Sebastian & Weiß, Christian H., 2014. "Compound Poisson INAR(1) processes: Stochastic properties and testing for overdispersion," Computational Statistics & Data Analysis, Elsevier, vol. 77(C), pages 267-284.
    2. Sebastian Schweer, 2016. "A Goodness-of-Fit Test for Integer-Valued Autoregressive Processes," Journal of Time Series Analysis, Wiley Blackwell, vol. 37(1), pages 77-98, January.
    3. Sebastian Schweer & Christian H. Weiß, 2016. "Testing for Poisson arrivals in INAR(1) processes," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 25(3), pages 503-524, September.
    4. Christian H. Weiß & Annika Homburg & Pedro Puig, 2019. "Testing for zero inflation and overdispersion in INAR(1) models," Statistical Papers, Springer, vol. 60(3), pages 823-848, June.
    5. Šárka Hudecová & Marie Hušková & Simos G. Meintanis, 2021. "Goodness–of–Fit Tests for Bivariate Time Series of Counts," Econometrics, MDPI, vol. 9(1), pages 1-20, March.
    6. Simos Meintanis & Dimitris Karlis, 2014. "Validation tests for the innovation distribution in INAR time series models," Computational Statistics, Springer, vol. 29(5), pages 1221-1241, October.
    7. Christian Gouriéroux & Yang Lu, 2019. "Negative Binomial Autoregressive Process with Stochastic Intensity," Journal of Time Series Analysis, Wiley Blackwell, vol. 40(2), pages 225-247, March.
    8. Boris Aleksandrov & Christian H. Weiß & Carsten Jentsch, 2022. "Goodness‐of‐fit tests for Poisson count time series based on the Stein–Chen identity," Statistica Neerlandica, Netherlands Society for Statistics and Operations Research, vol. 76(1), pages 35-64, February.
    9. Puig, Pedro & Weiß, Christian H., 2020. "Some goodness-of-fit tests for the Poisson distribution with applications in Biodosimetry," Computational Statistics & Data Analysis, Elsevier, vol. 144(C).
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