Absolute regularity and ergodicity of Poisson count processes

By Michael H. Neumann. 2011

172 citations recorded by the source corpus.

We consider a class of observation-driven Poisson count processes where the current value of the accompanying intensity process depends on previous values of both processes. We show under a contractive condition that the bivariate process has a unique stationary distribution and that the stationary version of the count process is absolutely regular. Moreover, since the intensities can be written as measurable functionals of the count variables we conclude that the bivariate process is ergodic. As an important application of these results, we show how a test method previously used in the case of independent Poisson data can be used in the case of Poisson count processes.

Subfields: Artificial Intelligence, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/52c5947f990419c9eec0226e4ba6e1b7539e6d9f

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Citations in the corpus

Degenerate $$U$$- and $$V$$-statistics under ergodicity: asymptotics, bootstrap and applications in statistics (2013, methodology): “According to Theorem 2.1(i) and Theorem 3.1(iii) in Neumann (2011), the contractive condition (G3)(i) below ensures the existence of a unique strictly stationary and ergodic solution ((Yt , λt )′)t∈Z to the system of model equations above.”
Self-Excited Threshold Poisson Autoregression (2013, background): “ and it is relatively easy to ﬁt via maximum likelihood. Fokianos et al. (2009) studied the model and established the asymptotic theory of the parameter estimates by introducing a small perturbation. Neumann (2011) considered some contracting dynamics of λt and derived mixing condition of the count ”
Theory and Inference for a Class of Observation-driven Models with Application to Time Series of Counts (2012, methodology): “…author: Department of Statistics, Columbia University, 1255 Amsterdam Avenue, MC4690, New York, NY 10027, USA; Email: hengliu@stat.columbia.edu Later Neumann (2011) relaxed the linear assumption to a general contracting evolution rule and proved the absolute regularity for this Poisson count…”
Poisson QMLE of Count Time Series Models (2016, background): “To tackle these problems, Fokianos and Tjøstheim (2011) proposed a model in which the conditional distribution of Xt given its past values is Poisson with intensity parameter t D e t , where t D !”
Theory and inference for a class of nonlinear models with application to time series of counts (2016, background): “Neumann (2011) relaxed the linear assumption on the Poisson INGARCH to a general contracting evolution rule and proved the absolute regularity for this Poisson count process, and Doukhan, Fokianos, and Tjøstheim (2012) showed the existence of moments under similar conditions by utilizing the…”
Changepoints in times series of counts (2012, background): “Fokianos et al. (2009), Neumann (2011) and Franke (2010) showed various kinds of weak dependence properties for Poisson autoregressions.”
Parameter change test for zero-inflated generalized Poisson autoregressive models (2016, result): “Based on the results of Neumann,[37] one can show the stationarity and ergodicity of the process under assumption (A1).”
Ergodicity of observation-driven time series models and consistency of the maximum likelihood estimator (2012, methodology): “The approach developed by Neumann (2011) is more direct but is based on a contraction assumption on the intensity of the Poisson variable which is not satisfied, for example neither in the log-linear Poisson autoregression model nor in the Poisson threshold model.”
Count Time Series: A Methodological Review (2021, background): “Series properties are established in Neumann (2011), Doukhan, Fokianos, and Tjøstheim (2012), and Douc, Doukhan, and Moulines (2013).”
Count Time Series Models (2012, methodology): “To complement the presentation, ergodicity of model (22) has been proved by employing the contraction assumption (30) for p = q = 1 on f (·) , see Fokianos and Tjøstheim (2010) (for its perturbed version), and Neumann (2011), Franke (2010) for the response process.”
QUASI‐LIKELIHOOD INFERENCE FOR NEGATIVE BINOMIAL TIME SERIES MODELS (2013, background): “/ in (3), see also Fokianos et al. (2009a), Neumann (2011) and Fokianos and Tjøstheim (2012).”
Parameter Change Test for Poisson Autoregressive Models (2014, background): “Neumann (2011) established that ¹.”