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

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