Inference for nonstationary time series of counts with application to change-point problems

We consider an integer-valued time series $Y=(Y_t)_{t\in\Z}$ where the models after a time $k^*$ is Poisson autoregressive with the conditional mean that depends on a parameter $\theta^*\in\Theta\subset\R^d$. The structure of the process before $k^*$ is unknown;? it could be any other integer-valued time series, that is, the process $Y$ could be nonstationary.? It is established that the maximum likelihood estimator of $\theta^*$ computed on the nonstationary observations is consistent and asymptotically normal. Next, we carry out the sequential change-point detection in a large class of Poisson autoregressive models. We propose a monitoring scheme for detecting change in the model. The procedure is based on an updated estimator which is computed without the historical observations. The asymptotic behavior of the detector is studied, in particular, the above result on the inference in a nonstationary setting are applied to prove that the proposed procedure is consistent. A simulation study as well as a real data application are provided.