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Improving Ratio Estimators of Second Order Point Process Characteristics

Author

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  • Dietrich Stoyan
  • Helga Stoyan

Abstract

Ripley's K function, the L function and the pair correlation function are important second order characteristics of spatial point processes. These functions are usually estimated by ratio estimators, where the numerators are Horvitz–Thompson edge corrected estimators and the denominators estimate the intensity or its square. It is possible to improve these estimators with respect to bias and estimation variance by means of adapted distance dependent intensity estimators. Further improvement is possible by means of refined estimators of the square of intensity. All this is shown by statistical analysis of simulated Poisson, cluster and hard core processes.

Suggested Citation

  • 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.
    • Handle: RePEc:bla:scjsta:v:27:y:2000:i:4:p:641-656
    DOI: 10.1111/1467-9469.00213
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    Citations

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    Cited by:

    1. Tscheschel, André & Chiu, Sung Nok, 2008. "Quasi-plus sampling edge correction for spatial point patterns," Computational Statistics & Data Analysis, Elsevier, vol. 52(12), pages 5287-5295, August.
    2. 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.
    3. 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.
    4. Eric Marcon & Florence Puech, 2016. "A typology of distance-based measures of spatial concentration," Post-Print halshs-00679993, HAL.
    5. 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.
    6. 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.
    7. Eric Marcon & Florence Puech, 2012. "A typology of distance-based measures of spatial concentration," Working Papers halshs-00679993, HAL.
    8. Guan, Yongtao, 2007. "A least-squares cross-validation bandwidth selection approach in pair correlation function estimations," Statistics & Probability Letters, Elsevier, vol. 77(18), pages 1722-1729, December.
    9. 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.
    10. Yange Wang & Xiaohui Yang & Zhongjie Shi, 2013. "The Formation of the Patterns of Desert Shrub Communities on the Western Ordos Plateau, China: The Roles of Seed Dispersal and Sand Burial," PLOS ONE, Public Library of Science, vol. 8(7), pages 1-9, July.
    11. M. Lieshout, 2006. "A J-Function for Marked Point Patterns," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 58(2), pages 235-259, June.
    12. 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.
    13. Lothar Heinrich & Michaela Prokešová, 2010. "On Estimating the Asymptotic Variance of Stationary Point Processes," Methodology and Computing in Applied Probability, Springer, vol. 12(3), pages 451-471, September.
    14. Xiao Wang & Bo Zhang & Kebin Zhang & Jinxing Zhou & Bilal Ahmad, 2015. "The Spatial Pattern and Interactions of Woody Plants on the Temperate Savanna of Inner Mongolia, China: The Effects of Alternating Seasonal Grazing-Mowing Regimes," PLOS ONE, Public Library of Science, vol. 10(7), pages 1-17, July.
    15. 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.
    16. 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.

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