Asymptotic Palm likelihood theory for stationary point processes

  • Michaela ProkešováEmail author
  • Eva B. Vedel Jensen


In the present paper, we propose a Palm likelihood approach as a general estimating principle for stationary point processes in \(\mathbf{R}^d\) for which the density of the second-order factorial moment measure is available in closed form or in an integral representation. Examples of such point processes include the Neyman–Scott processes and the log Gaussian Cox processes. The computations involved in determining the Palm likelihood estimator are simple. Conditions are provided under which the Palm likelihood estimator is strongly consistent and asymptotically normally distributed.


Asymptotic normality Cluster processes Consistency  Neyman–Scott processes Log Gaussian Cox processes Palm likelihood  Spatial point process Strong mixing 



This work was supported by projects GAČR 201/08/P100 and 201/10/0472 from the Czech Science Foundation and by Centre for Stochastic Geometry and Advanced Bioimaging, funded by the Villum Foundation.


  1. Baddeley, A. J., Turner, R. (2000). Practical maximum pseudolikelihood for spatial point processes. Australian& New Zealand Journal of Statistics, 42, 283–322.Google Scholar
  2. Bolthausen, E. (1982). On the central limit theorem for stationary mixing random fields. Annals of Probability, 10, 1047–1050.Google Scholar
  3. Crowder, M. J. (1986). On consistency and inconsistency of estimating equations. Econometric Theory, 2, 305–330.Google Scholar
  4. Daley, D. J., Vere-Jones, D. (2003). An Introduction to the Theory of Point Processes (2nd ed.). New York: Springer.Google Scholar
  5. Davidson, J. (1994). Stochastic Limit Theory. New York: Oxford University Press.Google Scholar
  6. Diggle, P. J. (2003). Statistical Analysis of Spatial Point Patterns. New York: Oxford University Press.Google Scholar
  7. Doukhan, P. (1994). Mixing: Properties and Examples. New York: Springer.Google Scholar
  8. Guan, Y. (2006). A composite likelihood approach in fitting spatial point process models. Journal of the American Statistical Association, 101, 1502–1512.Google Scholar
  9. Guan, Y., Sherman, M. (2007). On least squares fitting for stationary spatial point processes. Journal of the Royal Statistical Society: Series B, 69, 31–49.Google Scholar
  10. Guan, Y., Sherman, M., Calvin, J. A. (2007). On asymptotic properties of the mark variogram estimator of a marked point process. Journal of Statistical Planning and Inference, 137, 148–161.Google Scholar
  11. Guyon, X. (1995). Random Fields on a Network. New York: Springer.Google Scholar
  12. Heagerty, P. J., Lumley, T. (2000). Window subsampling of estimating functions with application to regression models. Journal of the American Statistical Association, 95, 197–211.Google Scholar
  13. Heinrich, L. (1988). Asymptotic Gaussianity of some estimators for reduced factorial moment measures and product densities of stationary poisson cluster processes. Statistics, 19, 87–106.Google Scholar
  14. Heinrich, L. (1992). Minimum contrast estimates for parameters of spatial ergodic point processes. In Transactions of the 11th Prague Conference on Random Processes, Information Theory and Statistical Decision Functions. Prague: Academic Publishing House.Google Scholar
  15. Heinrich, L. (2012). Asymptotic methods in statistics of random point processes. In E. Spodarev (Ed.), Lecture notes on Stochastic Geometry, Spatial Statistics and Random Fields. New York: Springer (to appear).Google Scholar
  16. Ibragimov, I. A., Linnik, Y. V. (1971). Independent and Stationary Sequences of Random Variables. Groningen: Wolters-Noordhoff.Google Scholar
  17. Illian, J., Penttinen, A., Stoyan, H., Stoyan, D. (2008). Statistical Analysis and Modelling of Spatial Point Patterns. Chichester: Wiley.Google Scholar
  18. Jolivet, E. (1981). Central limit theorem and convergence of empirical processes of stationary point processes. In P. Bartfai, J. Tomko (Eds.), Point Processes and Queueing Problems. Amsterdam: North-Holland.Google Scholar
  19. Jonsdottir, K. Y., Rønn-Nielsen, A., Mouridsen, K., Jensen, E. B. V. (2011). Lévy based modelling in brain imaging. In C. S. G. B. Research (Ed.), Report (pp. 11–2). Aarhus: Department of Mathematics, Aarhus University (Submitted).Google Scholar
  20. Lindsay, B. G. (1988). Composite likelihood methods. Contemporary Mathematics, 80, 221–239.Google Scholar
  21. Møller, J., Waagepetersen, R. P. (2003). Statistical Inference and Simulation for Spatial Point Processes. Boca Raton: Chapman& Hall/CRC.Google Scholar
  22. Møller, J., Waagepetersen, R. P. (2007). Modern statistics for spatial point processes. Scandinavian Journal of Statistics, 34, 643–684.Google Scholar
  23. Møller, J., Syversveen, A. R., Waagepetersen, R. P. (1998). Log Gaussian Cox processes. Scandinavian Journal of Statistics, 25, 451–482.Google Scholar
  24. Nelder, J. A., Mead, R. (1965). A simplex method for function minimization. Computer Journal, 7, 308.Google Scholar
  25. Ogata, Y., Katsura, K. (1991). Maximum likelihood estimates of fractal dimension for random spatial patterns. Biometrika, 78, 463–474.Google Scholar
  26. Politis, D. N., Sherman, M. (2001). Moment estimation for statistics from marked point processes. Journal of the Royal Statistical Society: Series B, 63, 261–275.Google Scholar
  27. Rosenblatt, M. (1956). A central limit theorem and a strong mixing condition. Proceedings of the National Academy of Sciences, 42, 43–47.Google Scholar
  28. Rue, H., Martino, S., Chopin, N. (2009). Approximate Bayesian inference for latent Gaussian models using integrated nested Laplace approximations (with discussion). Journal of the Royal Statistical Society: Series B, 71, 319–392.Google Scholar
  29. Stein, M. L. (1999). Interpolation of Spatial Data. New York: Springer.Google Scholar
  30. Stoyan, D., Kendall, W. S., Mecke, J. (1995). Stochastic Geometry and its Applications (2nd ed.). Chichester: Wiley.Google Scholar
  31. Tanaka, U., Ogata, Y., Stoyan, D. (2008). Parameter Estimation and Model Selection for Neyman–Scott Point Processes. Biometrical Journal, 50, 43–57.Google Scholar
  32. Thomas, M. (1949). A generalization of Poisson’s binomial limit for use in ecology. Biometrika, 36, 18–25.Google Scholar
  33. Waagepetersen, R. P. (2007). An estimating function approach to inference for inhomogeneous Neyman–Scott processes. Biometrics, 63, 252–258.Google Scholar
  34. Wills, J. M. (1970). Zum Verhältnis von Volumen zu Oberfläche bei Convexen Körpern. Archiv der Mathematik, 21, 557–560.Google Scholar
  35. Zhengyan, L., Chuanrong, L. (1996). Limit Theory for Mixing Dependent Random Variables. Dordrecht: Kluwer Academic Publishers.Google Scholar

Copyright information

© The Institute of Statistical Mathematics, Tokyo 2012

Authors and Affiliations

  1. 1.Department of Probability and Statistics, Faculty of Mathematics and PhysicsCharles UniversityPragueCzech Republic
  2. 2.Department of Mathematics, Centre for Stochastic Geometry and Advanced Bioimaging Aarhus UniversityAarhus CDenmark

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