IDEAS home Printed from https://ideas.repec.org/a/spr/sistpr/v17y2014i2p121-138.html
   My bibliography  Save this article

Central limit theorems for empirical product densities of stationary point processes

Author

Listed:
  • Lothar Heinrich
  • Stella Klein

Abstract

We prove the asymptotic normality of kernel estimators of second- and higher-order product densities (and of the pair correlation function) for spatially homogeneous (and isotropic) point processes observed on a sampling window $$W_n$$ W n , which is assumed to expand unboundedly in all directions as $$n \rightarrow \infty \,$$ n → ∞ . We first study the asymptotic behavior of the covariances of the empirical product densities under minimal moment and weak dependence assumptions. The proof of the main results is based on the Brillinger-mixing property of the underlying point process and certain smoothness conditions on the higher-order reduced cumulant measures. Finally, the obtained limit theorems enable us to construct $$\chi ^2$$ χ 2 -goodness-of-fit tests for hypothetical product densities. Copyright Springer Science+Business Media Dordrecht 2014

Suggested Citation

  • Lothar Heinrich & Stella Klein, 2014. "Central limit theorems for empirical product densities of stationary point processes," Statistical Inference for Stochastic Processes, Springer, vol. 17(2), pages 121-138, July.
    • Handle: RePEc:spr:sistpr:v:17:y:2014:i:2:p:121-138
    DOI: 10.1007/s11203-014-9094-5
    as

    Download full text from publisher

    File URL: http://hdl.handle.net/10.1007/s11203-014-9094-5
    Download Restriction: Access to full text is restricted to subscribers.
    File URL: https://libkey.io/10.1007/s11203-014-9094-5?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. E. Jolivet, 1984. "Upper bound of the speed of convergence of moment density Estimators for stationary point processes," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 31(1), pages 349-360, December.
    2. 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.
    3. Dietrich Stoyan & Helga Stoyan, 2000. "Improving Ratio Estimators of Second Order Point Process Characteristics," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 27(4), pages 641-656, December.
    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. Heinrich, Lothar, 2018. "Brillinger-mixing point processes need not to be ergodic," Statistics & Probability Letters, Elsevier, vol. 138(C), pages 31-35.

    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. Heinrich Lothar & Klein Stella, 2011. "Central limit theorem for the integrated squared error of the empirical second-order product density and goodness-of-fit tests for stationary point processes," Statistics & Risk Modeling, De Gruyter, vol. 28(4), pages 359-387, December.
    2. Jesper Møller & Carlos Díaz‐Avalos, 2010. "Structured Spatio‐Temporal Shot‐Noise Cox Point Process Models, with a View to Modelling Forest Fires," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 37(1), pages 2-25, March.
    3. Tonglin Zhang & Ge Lin, 2009. "Cluster Detection Based on Spatial Associations and Iterated Residuals in Generalized Linear Mixed Models," Biometrics, The International Biometric Society, vol. 65(2), pages 353-360, June.
    4. Eric Marcon & Florence Puech, 2012. "A typology of distance-based measures of spatial concentration," Working Papers halshs-00679993, HAL.
    5. 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).
    6. Marcon, Eric & Puech, Florence, 2017. "A typology of distance-based measures of spatial concentration," Regional Science and Urban Economics, Elsevier, vol. 62(C), pages 56-67.
    7. 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.
    8. 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.
    9. 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.
    10. Coeurjolly, Jean-François, 2015. "Almost sure behavior of functionals of stationary Gibbs point processes," Statistics & Probability Letters, Elsevier, vol. 97(C), pages 241-246.
    11. 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.
    12. Tonglin Zhang & Ge Lin, 2008. "Identification of local clusters for count data: a model-based Moran's I test," Journal of Applied Statistics, Taylor & Francis Journals, vol. 35(3), pages 293-306.
    13. 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.
    14. Shaochuan Lu, 2012. "Markov modulated Poisson process associated with state-dependent marks and its applications to the deep earthquakes," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 64(1), pages 87-106, February.
    15. Daniel, Jeffrey & Horrocks, Julie & Umphrey, Gary J., 2018. "Penalized composite likelihoods for inhomogeneous Gibbs point process models," Computational Statistics & Data Analysis, Elsevier, vol. 124(C), pages 104-116.
    16. Jonatan A. González & Francisco J. Rodríguez-Cortés & Elvira Romano & Jorge Mateu, 2021. "Classification of Events Using Local Pair Correlation Functions for Spatial Point Patterns," Journal of Agricultural, Biological and Environmental Statistics, Springer;The International Biometric Society;American Statistical Association, vol. 26(4), pages 538-559, December.
    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. Mohammad Ghorbani & Ottmar Cronie & Jorge Mateu & Jun Yu, 2021. "Functional marked point processes: a natural structure to unify spatio-temporal frameworks and to analyse dependent functional data," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 30(3), pages 529-568, September.
    19. Giada Adelfio & Frederic Schoenberg, 2009. "Point process diagnostics based on weighted second-order statistics and their asymptotic properties," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 61(4), pages 929-948, December.
    20. A. Baddeley & J. Møller & A. Pakes, 2008. "Properties of residuals for spatial point processes," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 60(3), pages 627-649, September.

    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:sistpr:v:17:y:2014:i:2:p:121-138. 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.