  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.
