Inference and testing for structural change in time series of counts model

We consider here together the inference questions and the change-point problem in Poisson autoregressions (see Tj{\o}stheim, 2012). 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 is shown that the conditional mean can be written as a function of lagged observations. In 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 $\theta_0$. We propose two tests based on the likelihood of the observations. Under the null hypothesis (i.e. no change), it is proved that both those test statistics converge to an explicit distribution. Consistencies under alternatives are proved for both tests. Simulation results show how those procedure work practically, and an application to real data is also processed.