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

TL;DR

This paper extends the Campbell equilibrium equation to non-hereditary Gibbs point processes, introduces the concept of removable points, and proposes a consistent two-step pseudo-likelihood estimation method for model parameters.

Contribution

It generalizes the Campbell equilibrium equation to non-hereditary processes and develops a new estimation procedure involving removable points.

Findings

Proved the consistency of the estimation procedure.

Extended the equilibrium equation to non-hereditary settings.

Validated the method for both hereditary and non-hereditary models.

Abstract

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 [Math. Nachr. 88 (1979) 105--115], to the non-hereditary setting and consequently introduce 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 estimate the hardcore parameter, then estimate 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 non-hereditary settings.