Estimation of the intensity parameter of the germ-grain Quermass-interaction model when the number of germs is not observed

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

1 LPP - Laboratoire Paul Painlevé - UMR 8524
2 Probabilités, statistique et calcul scientifique
3 Department of Mathematics, Faculty of Electrical Engineering

The Quermass-interaction model allows to generalise the classical germ-grain Boolean model in adding a morphological interaction between the grains. It enables to model random structures with specific morphologies which are unlikely to be generated from a Boolean model. The Quermass-interaction model depends in particular on an intensity parameter, which is impossible to estimate from classical likelihood or pseudo-likelihood approaches because the number of points is not observable from a germ-grain set. In this paper, we present a procedure based on the Takacs-Fiksel method which is able to estimate all parameters of the Quermass-interaction model, including the intensity. An intensive simulation study is conducted to assess the efficiency of the procedure and to provide practical recommendations. It also illustrates that the estimation of the intensity parameter is crucial in order to identify the model. The Quermass-interaction model is finally fitted by our method to P. Diggle's heather dataset.

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

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

David Dereudre, Frédéric Lavancier, Katerina Helisova Stankova. Estimation of the intensity parameter of the germ-grain Quermass-interaction model when the number of germs is not observed. Scandinavian Journal of Statistics, 2014, pp.1-21. ⟨10.1111/sjos.12064⟩. ⟨hal-00720744v3⟩