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

TL;DR

This paper introduces a Takacs-Fiksel based method to estimate the intensity parameter of the Quermass-interaction model, which generalizes germ-grain models with morphological interactions, especially when the number of germs is unobserved.

Contribution

It presents a novel estimation procedure for all parameters of the Quermass-interaction model, including the intensity, using the Takacs-Fiksel method, validated through extensive simulations.

Findings

The method accurately estimates the intensity parameter.

Estimating the intensity is key for model identification.

Application to real data demonstrates practical utility.

Abstract

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.