Campbell equilibrium equation and pseudo-likelihood estimation for non-hereditary Gibbs point processes

In this paper, we study Gibbs point processes involving a hardcore interaction which is not necessarily hereditary. We first extend the famous Campbell equilibrium equation, initially proposed by Nguyen and Zessin (\cite{NZ}), to the non-hereditary setting and introduce consequently the new concept of removable points. A modified version of the pseudo-likelihood estimator is then proposed, which involves these removable points. We consider the following two-step estimation procedure: First estimating the hardcore parameter, then estimating the smooth interaction parameter by pseudo-likelihood, where the hardcore parameter estimator is plugged in. We prove the consistency of this procedure in both the hereditary and the non-hereditary setting.