  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.
