We will review some results about the statistical inference of the branching rate of certain piecewise deterministic Markov models. Whereas their abstract statistical structure is relatively well-known from a parametric point of view, some recent applications (arising for instance from cell division models in biology) have renewed the interest of such statistical models, in particular from a non-parametric and testing point of view. In that context, new difficulties emerge, in particular from the perspective of implementing procedures. We will present some generic inference results (including a real-data study on Escherichia Coli) and explain how fragile the information is with respect to the observation scheme (namely observing data in a stationary regime, at branching times or simply the whole genealogy over a given fixed time), a point that is sometimes overlooked by practitioners.
