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Likelihood Inference for Unions of Interacting Discs

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  • JESPER MØLLER
  • KATEŘINA HELISOVÁ

Abstract

. 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.

Suggested Citation

  • Jesper Møller & Kateřina Helisová, 2010. "Likelihood Inference for Unions of Interacting Discs," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 37(3), pages 365-381, September.
    • Handle: RePEc:bla:scjsta:v:37:y:2010:i:3:p:365-381
    DOI: 10.1111/j.1467-9469.2009.00660.x
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    Cited by:

    1. Kateřina Staňková Helisová & Jakub Staněk, 2014. "Dimension Reduction in Extended Quermass-Interaction Process," Methodology and Computing in Applied Probability, Springer, vol. 16(2), pages 355-368, June.
    2. Vesna Gotovac Dogaš & Kateřina Helisová, 2021. "Testing Equality of Distributions of Random Convex Compact Sets via Theory of 𝕹 $\mathfrak {N}$ -Distances," Methodology and Computing in Applied Probability, Springer, vol. 23(2), pages 503-526, June.
    3. Johan Debayle & Vesna Gotovac Ðogaš & Kateřina Helisová & Jakub Staněk & Markéta Zikmundová, 2021. "Assessing Similarity of Random sets via Skeletons," Methodology and Computing in Applied Probability, Springer, vol. 23(2), pages 471-490, June.
    4. Markéta Zikmundová & Kateřina Staňková Helisová & Viktor Beneš, 2014. "On the Use of Particle Markov Chain Monte Carlo in Parameter Estimation of Space-Time Interacting Discs," Methodology and Computing in Applied Probability, Springer, vol. 16(2), pages 451-463, June.
    5. Viktor Beneš & Jakub Večeřa & Milan Pultar, 2019. "Planar Segment Processes with Reference Mark Distributions, Modeling and Estimation," Methodology and Computing in Applied Probability, Springer, vol. 21(3), pages 683-698, September.
    6. Markéta Zikmundová & Kateřina Staňková Helisová & Viktor Beneš, 2012. "Spatio-Temporal Model for a Random Set Given by a Union of Interacting Discs," Methodology and Computing in Applied Probability, Springer, vol. 14(3), pages 883-894, September.
    7. Athanasios C. Micheas & Jiaxun Chen, 2018. "sppmix: Poisson point process modeling using normal mixture models," Computational Statistics, Springer, vol. 33(4), pages 1767-1798, December.

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