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

Boris Aleksandrov, Christian H. Weiß (), Simon Nik, Maxime Faymonville and Carsten Jentsch
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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

Metrika: International Journal for Theoretical and Applied Statistics, 2024, vol. 87, issue 7, No 4, 843-887

Abstract: 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.

Keywords: Bivariate negative-binomial distribution; Count time series; Diagnostic tests; Iterated thinning; NB-IINAR(1) model; Stein identity; 60G10; 62F03; 62F05; 62M10 (search for similar items in EconPapers)
Date: 2024
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Citations: View citations in EconPapers (1)

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DOI: 10.1007/s00184-023-00934-0

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