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Planar Segment Processes with Reference Mark Distributions, Modeling and Estimation

Author

Listed:
  • Viktor Beneš

    (Charles University, Faculty of Mathematics and Physics, Department of Probability and Mathematical Statistics)

  • Jakub Večeřa

    (Charles University, Faculty of Mathematics and Physics, Department of Probability and Mathematical Statistics)

  • Milan Pultar

    (Charles University, Faculty of Mathematics and Physics, Department of Probability and Mathematical Statistics)

Abstract

The paper deals with planar segment processes given by a density with respect to the Poisson process. Parametric models involve reference distributions of directions and/or lengths of segments. These distributions generally do not coincide with the corresponding observed distributions. Statistical methods are presented which first estimate scalar parameters by known approaches and then the reference distribution is estimated non-parametrically. Besides a general theory we offer two models, first a Gibbs type segment process with reference directional distribution and secondly an inhomogeneous process with reference length distribution. The estimation is demonstrated in simulation studies where the variability of estimators is presented graphically.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:metcap:v:21:y:2019:i:3:d:10.1007_s11009-017-9608-x
    DOI: 10.1007/s11009-017-9608-x
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    References listed on IDEAS

    as
    1. Jakub Večeřa & Viktor Beneš, 2016. "Interaction Processes for Unions of Facets, the Asymptotic Behaviour with Increasing Intensity," Methodology and Computing in Applied Probability, Springer, vol. 18(4), pages 1217-1239, December.
    2. 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.
    3. Chen, Song Xi, 1999. "Beta kernel estimators for density functions," Computational Statistics & Data Analysis, Elsevier, vol. 31(2), pages 131-145, August.
    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.
    5. K. V. Mardia, 1999. "Directional statistics and shape analysis," Journal of Applied Statistics, Taylor & Francis Journals, vol. 26(8), pages 949-957.
    6. Večeřa, Jakub & Beneš, Viktor, 2017. "Approaches to asymptotics for U-statistics of Gibbs facet processes," Statistics & Probability Letters, Elsevier, vol. 122(C), pages 51-57.
    7. 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.
    8. Zbyněk Pawlas, 2014. "Self-crossing Points of a Line Segment Process," Methodology and Computing in Applied Probability, Springer, vol. 16(2), pages 295-309, June.
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