Takacs–Fiksel Method for Stationary Marked Gibbs Point Processes


Jean-François Coeurjolly, 51 rue des Mathématiques – Campus de Saint Martin d'Hères, Saint Martin d'Hères 38402, France.
E-mail: jean-francois.coeurjolly@upmf-grenoble.fr


Abstract.  This article studies a method to estimate the parameters governing the distribution of a stationary marked Gibbs point process. This procedure, known as the Takacs–Fiksel method, is based on the estimation of the left and right hand sides of the Georgii–Nguyen–Zessin formula and leads to a family of estimators due to the possible choices of test functions. We propose several examples illustrating the interest and flexibility of this procedure. We also provide sufficient conditions based on the model and the test functions to derive asymptotic properties (consistency and asymptotic normality) of the resulting estimator. The different assumptions are discussed for exponential family models and for a large class of test functions. A short simulation study is proposed to assess the correctness of the methodology and the asymptotic results.