Doukhan, Paul; Fokianos, Konstantinos; Rynkiewicz, Joseph Mixtures of nonlinear Poisson autoregressions. (English) Zbl 1468.62334 J. Time Ser. Anal. 42, No. 1, 107-135 (2021). Summary: We study nonlinear infinite order Markov switching integer-valued ARCH models for count time series data. Markov switching models take into account complex dynamics and can deal with several stylistic facts of count data including proper modelling of nonlinearities, overdispersion and outliers. We study structural properties of those models. Under mild conditions, we prove consistency and asymptotic normality of the maximum likelihood estimator for the case of finite order autoregression. In addition, we give conditions which imply that the marginal likelihood ratio test, for testing the number of regimes, converges to a Gaussian process. This result enables us to prove that the BIC provides a consistent estimator for selecting the true number of regimes. A real data example illustrates the methodology and compares this approach with alternative methods. Cited in 4 Documents MSC: 62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH) 62B10 Statistical aspects of information-theoretic topics 60G15 Gaussian processes 60B12 Limit theorems for vector-valued random variables (infinite-dimensional case) Keywords:Bayesian information criterion; ergodicity; identifiability; nonlinear time series; penalized likelihood; stationarity PDFBibTeX XMLCite \textit{P. Doukhan} et al., J. Time Ser. Anal. 42, No. 1, 107--135 (2021; Zbl 1468.62334) Full Text: DOI Link References: [1] AhmadA, FranqC. 2016. Poisson QMLE of count time series models. Journal of Time Series Analysis37: 291-314. · Zbl 1381.62244 [2] AlbertPS. 1991. A two-state Markov mixture model for a time series of epileptic seizure counts. Biometrics47: 1371-1381. [3] BazziM, BlasquesF, KoopmanSJ, LucasA. 2017. 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