Fitting all parameters of the Quermass model by the Takacs-Fiksel method

David Dereudre (1) , Frédéric Lavancier (2) , Katerina Helisova Stankova (3)

1 LPP - Laboratoire Paul Painlevé - UMR 8524
2 LMJL - Laboratoire de Mathématiques Jean Leray
3 Department of Mathematics, Faculty of Electrical Engineering

The Quermass model is a generalization of the classical germ-grain Boolean model for which a morphological interaction is added. It allows to model random structures with specific morphologies which are unlikely to come from a Boolean model. The Quermass model depends on three interacting parameters and on an intensity parameter. Since the number of points is not observable from a germ-grain set, the estimation of all parameters is not possible from classical likelihood or pseudo-likelihood approaches. In this paper, we present a procedure based on the Takacs-Fiksel method which is able to fit all parameters of the Quermass model, included the intensity. An intensive simulation study is conducted to assess the efficiency of the procedure and to provide practical recommendations. An application to heather data, initially studied by P. Diggle, is finally proposed.

Mots clés: Gibbs Point Process; germ-grain model; Quermass-Interaction Process; Area-Process; Perimeter-Process; Takacs-Fiksel estimator.; Takacs-Fiksel estimator

Domaines: Statistiques [math.ST]; Théorie [stat.TH]

David Dereudre, Frédéric Lavancier, Katerina Helisova Stankova. Fitting all parameters of the Quermass model by the Takacs-Fiksel method. 2012. ⟨hal-00720744v1⟩