IDEAS home Printed from https://ideas.repec.org/a/spr/aistmt/v69y2017i2d10.1007_s10463-015-0536-7.html
   My bibliography  Save this article

Median-based estimation of the intensity of a spatial point process

Author

Listed:
  • Jean-François Coeurjolly

    (Univ. Grenoble Alpes)

Abstract

This paper is concerned with a robust estimator of the intensity of a stationary spatial point process. The estimator corresponds to the median of a jittered sample of the number of points, computed from a tessellation of the observation domain. We show that this median-based estimator satisfies a Bahadur representation from which we deduce its consistency and asymptotic normality under mild assumptions on the spatial point process. Through a simulation study, we compare the new estimator, in particular, with the standard one counting the mean number of points per unit volume. The empirical study confirms the asymptotic properties established in the theoretical part and shows that the median-based estimator is more robust to outliers than standard procedures.

Suggested Citation

  • Jean-François Coeurjolly, 2017. "Median-based estimation of the intensity of a spatial point process," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 69(2), pages 303-331, April.
    • Handle: RePEc:spr:aistmt:v:69:y:2017:i:2:d:10.1007_s10463-015-0536-7
    DOI: 10.1007/s10463-015-0536-7
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10463-015-0536-7
    File Function: Abstract
    Download Restriction: Access to the full text of the articles in this series is restricted.
    File URL: https://libkey.io/10.1007/s10463-015-0536-7?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. Guan, Yongtao & Loh, Ji Meng, 2007. "A Thinned Block Bootstrap Variance Estimation Procedure for Inhomogeneous Spatial Point Patterns," Journal of the American Statistical Association, American Statistical Association, vol. 102, pages 1377-1386, December.
    2. Michaela Prokešová & Eva Jensen, 2013. "Asymptotic Palm likelihood theory for stationary point processes," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 65(2), pages 387-412, April.
    3. Jean-François Coeurjolly & Frédéric Lavancier, 2013. "Residuals and goodness-of-fit tests for stationary marked Gibbs point processes," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 75(2), pages 247-276, March.
    4. A. Baddeley & R. Turner & J. Møller & M. Hazelton, 2005. "Residual analysis for spatial point processes (with discussion)," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 67(5), pages 617-666, November.
    5. Vaart,A. W. van der, 2000. "Asymptotic Statistics," Cambridge Books, Cambridge University Press, number 9780521784504.
    6. Machado, Jose A.F. & Silva, J. M. C. Santos, 2005. "Quantiles for Counts," Journal of the American Statistical Association, American Statistical Association, vol. 100, pages 1226-1237, December.
    7. Tomáš Mrkvička & Ilya Molchanov, 2005. "Optimisation of linear unbiased intensity estimators for point processes," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 57(1), pages 71-81, March.
    8. Renato Assunção & Peter Guttorp, 1999. "Robustness for Inhomogeneous Poisson Point Processes," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 51(4), pages 657-678, December.
    9. J. A. Adell & P. Jodrá, 2005. "The median of the poisson distribution," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 61(3), pages 337-346, June.
    10. Rasmus Waagepetersen & Yongtao Guan, 2009. "Two‐step estimation for inhomogeneous spatial point processes," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 71(3), pages 685-702, June.
    11. Baddeley, Adrian & Turner, Rolf, 2005. "spatstat: An R Package for Analyzing Spatial Point Patterns," Journal of Statistical Software, Foundation for Open Access Statistics, vol. 12(i06).
    12. Yanyuan Ma & Marc Genton & Emanuel Parzen, 2011. "Asymptotic properties of sample quantiles of discrete distributions," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 63(2), pages 227-243, April.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Jean-François Coeurjolly & Joëlle Rousseau Trépanier, 2020. "The median of a jittered Poisson distribution," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 83(7), pages 837-851, October.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Michaela Prokešová & Jiří Dvořák & Eva B. Vedel Jensen, 2017. "Two-step estimation procedures for inhomogeneous shot-noise Cox processes," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 69(3), pages 513-542, June.
    2. Nicoletta D’Angelo & Marianna Siino & Antonino D’Alessandro & Giada Adelfio, 2022. "Local spatial log-Gaussian Cox processes for seismic data," AStA Advances in Statistical Analysis, Springer;German Statistical Society, vol. 106(4), pages 633-671, December.
    3. Yu Ryan Yue & Ji Meng Loh, 2011. "Bayesian Semiparametric Intensity Estimation for Inhomogeneous Spatial Point Processes," Biometrics, The International Biometric Society, vol. 67(3), pages 937-946, September.
    4. Ute Hahn & Eva B. Vedel Jensen, 2016. "Hidden Second-order Stationary Spatial Point Processes," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 43(2), pages 455-475, June.
    5. Victor Chernozhukov & Iván Fernández-Val & Blaise Melly & Kaspar Wüthrich, 2020. "Generic Inference on Quantile and Quantile Effect Functions for Discrete Outcomes," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 115(529), pages 123-137, January.
    6. Abdollah Jalilian, 2017. "Modelling and classification of species abundance: a case study in the Barro Colorado Island plot," Journal of Applied Statistics, Taylor & Francis Journals, vol. 44(13), pages 2401-2409, October.
    7. D'Angelo, Nicoletta & Adelfio, Giada & Mateu, Jorge, 2023. "Locally weighted minimum contrast estimation for spatio-temporal log-Gaussian Cox processes," Computational Statistics & Data Analysis, Elsevier, vol. 180(C).
    8. Andrew J Edelman, 2012. "Positive Interactions between Desert Granivores: Localized Facilitation of Harvester Ants by Kangaroo Rats," PLOS ONE, Public Library of Science, vol. 7(2), pages 1-9, February.
    9. Amanda S. Hering & Sean Bair, 2014. "Characterizing spatial and chronological target selection of serial offenders," Journal of the Royal Statistical Society Series C, Royal Statistical Society, vol. 63(1), pages 123-140, January.
    10. Ushio Tanaka & Yosihiko Ogata, 2014. "Identification and estimation of superposed Neyman–Scott spatial cluster processes," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 66(4), pages 687-702, August.
    11. Tilman M. Davies & Martin L. Hazelton, 2013. "Assessing minimum contrast parameter estimation for spatial and spatiotemporal log‐Gaussian Cox processes," Statistica Neerlandica, Netherlands Society for Statistics and Operations Research, vol. 67(4), pages 355-389, November.
    12. Laurent, Jean-Paul & Sestier, Michael & Thomas, Stéphane, 2016. "Trading book and credit risk: How fundamental is the Basel review?," Journal of Banking & Finance, Elsevier, vol. 73(C), pages 211-223.
    13. Coeurjolly, Jean-François, 2015. "Almost sure behavior of functionals of stationary Gibbs point processes," Statistics & Probability Letters, Elsevier, vol. 97(C), pages 241-246.
    14. Coeurjolly, Jean-François & Reynaud-Bouret, Patricia, 2019. "A concentration inequality for inhomogeneous Neyman–Scott point processes," Statistics & Probability Letters, Elsevier, vol. 148(C), pages 30-34.
    15. Devin S. Johnson & Jeffrey L. Laake & Jay M. Ver Hoef, 2010. "A Model-Based Approach for Making Ecological Inference from Distance Sampling Data," Biometrics, The International Biometric Society, vol. 66(1), pages 310-318, March.
    16. Christophe Ange Napoléon Biscio & Frédéric Lavancier, 2017. "Contrast Estimation for Parametric Stationary Determinantal Point Processes," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 44(1), pages 204-229, March.
    17. Kenneth A. Flagg & Andrew Hoegh & John J. Borkowski, 2020. "Modeling Partially Surveyed Point Process Data: Inferring Spatial Point Intensity of Geomagnetic Anomalies," Journal of Agricultural, Biological and Environmental Statistics, Springer;The International Biometric Society;American Statistical Association, vol. 25(2), pages 186-205, June.
    18. Frédéric Lavancier & Arnaud Poinas & Rasmus Waagepetersen, 2021. "Adaptive estimating function inference for nonstationary determinantal point processes," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 48(1), pages 87-107, March.
    19. Jean-François Coeurjolly & Joëlle Rousseau Trépanier, 2020. "The median of a jittered Poisson distribution," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 83(7), pages 837-851, October.
    20. Yongtao Guan, 2008. "Variance estimation for statistics computed from inhomogeneous spatial point processes," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 70(1), pages 175-190, February.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:spr:aistmt:v:69:y:2017:i:2:d:10.1007_s10463-015-0536-7. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.