Journal
SCANDINAVIAN JOURNAL OF STATISTICS
Volume 37, Issue 3, Pages 365-381Publisher
WILEY
DOI: 10.1111/j.1467-9469.2009.00660.x
Keywords
Boolean model; connected component Markov process; disc process; edge effects; germ-grain model; quermass-interaction process; random closed set; simulation-based maximum likelihood; spatial Markov property; summary statistics
Categories
Funding
- Danish Natural Science Research Council [272-06-0442]
- Czech Government [MSM6840770038]
- [IAA101120604]
- Villum Fonden [00008721] Funding Source: researchfish
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This is probably the first paper which discusses likelihood inference for a random set using a germ-grain model, where the individual grains are unobservable, edge effects occur and other complications appear. We consider the case where the grains form a disc process modelled by a marked point process, where the germs are the centres and the marks are the associated radii of the discs. We propose to use a recent parametric class of interacting disc process models, where the minimal sufficient statistic depends on various geometric properties of the random set, and the density is specified with respect to a given marked Poisson model (i.e. a Boolean model). We show how edge effects and other complications can be handled by considering a certain conditional likelihood. Our methodology is illustrated by analysing Peter Diggle's heather data set, where we discuss the results of simulation-based maximum likelihood inference and the effect of specifying different reference Poisson models.
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