Abstract
We consider here together the inference questions and the change-point problem in a large class of Poisson autoregressive models (see Tjøstheim, 2012 [34]). The conditional mean (or intensity) of the process is involved as a non-linear function of it past values and the past observations. Under Lipschitz-type conditions, it can be written as a function of lagged observations. For the latter model, assume that the link function depends on an unknown parameter $\theta_{0}$. The consistency and the asymptotic normality of the maximum likelihood estimator of the parameter are proved. These results are used to study change-point problem in the parameter $\theta_{0}$. From the likelihood of the observations, two tests are proposed. Under the null hypothesis (i.e. no change), each of these tests statistics converges to an explicit distribution. Consistencies under alternatives are proved for both tests. Simulation results show how those procedures work in practice, and applications to real data are also processed.
Highlights
Time series of counts appear as natural for modeling count events
Real advances have been made in count time series modeling during the last two decades
A model is characterized by the type of marginal distribution L(Yt/Ft−1), and the dependence structure between L(Yt/Ft−1) and the past
Summary
Time series of counts appear as natural for modeling count events. Some examples can be found in epidemiology (number of new infections), in finance (number of transactions per minute), in industrial quality control (number of defects), just to name a few. Douc et al (2013) [11] considered a class of observation-driven time series which covers linear, log-linear, and threshold Poisson autoregressions Their approach is based on a recent theory for Markov chains based upon Hairer and Mattingly (2006) [22] recent work; this allows existence and uniqueness of the invariant distribution for Markov chains without irreducibility. The properties of the general class of Poisson autoregressive model (1) have been investigated in Doukhan et al [13] Such infinite order processes provided a large way to take into account dependence on the past observations. The intervention problem studied by Fokianos and Fried [16, 17] is intended to sudden shift in the conditional mean of the process Such outlier could in some case be seen as a particular case of structural change problem that we develop here for a large class of models.
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