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Estimation of the Intensity Parameter of the Germ-Grain Quermass-Interaction Model when the Number of Germs is not Observed

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  • David Dereudre
  • Frédéric Lavancier
  • Kateřina Staňková Helisová

Abstract

type="main" xml:id="sjos12064-abs-0001"> The Quermass-interaction model allows to generalize 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 data set.

Suggested Citation

  • David Dereudre & Frédéric Lavancier & Kateřina Staňková Helisová, 2014. "Estimation of the Intensity Parameter of the Germ-Grain Quermass-Interaction Model when the Number of Germs is not Observed," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 41(3), pages 809-829, September.
    • Handle: RePEc:bla:scjsta:v:41:y:2014:i:3:p:809-829
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    File URL: http://hdl.handle.net/10.1111/sjos.12064
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    References listed on IDEAS

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    1. Nicolas Picard & Avner Bar‐Hen & Frédéric Mortier & Joël Chadœuf, 2009. "The Multi‐scale Marked Area‐interaction Point Process: A Model for the Spatial Pattern of Trees," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 36(1), pages 23-41, March.
    2. Schmidt, Volker & Spodarev, Evgueni, 2005. "Joint estimators for the specific intrinsic volumes of stationary random sets," Stochastic Processes and their Applications, Elsevier, vol. 115(6), pages 959-981, June.
    3. A. Baddeley & M. Lieshout, 1995. "Area-interaction point processes," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 47(4), pages 601-619, December.
    4. Jean-Franois Coeurjolly & David Dereudre & Rémy Drouilhet & Frédéric Lavancier, 2012. "Takacs–Fiksel Method for Stationary Marked Gibbs Point Processes," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 39(3), pages 416-443, September.
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