Abstract
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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Baddeley, A. J., Turner, R. (2000). Practical maximum pseudolikelihood for spatial point processes. Australian& New Zealand Journal of Statistics, 42, 283–322.
Bolthausen, E. (1982). On the central limit theorem for stationary mixing random fields. Annals of Probability, 10, 1047–1050.
Crowder, M. J. (1986). On consistency and inconsistency of estimating equations. Econometric Theory, 2, 305–330.
Daley, D. J., Vere-Jones, D. (2003). An Introduction to the Theory of Point Processes (2nd ed.). New York: Springer.
Davidson, J. (1994). Stochastic Limit Theory. New York: Oxford University Press.
Diggle, P. J. (2003). Statistical Analysis of Spatial Point Patterns. New York: Oxford University Press.
Doukhan, P. (1994). Mixing: Properties and Examples. New York: Springer.
Guan, Y. (2006). A composite likelihood approach in fitting spatial point process models. Journal of the American Statistical Association, 101, 1502–1512.
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.
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.
Guyon, X. (1995). Random Fields on a Network. New York: Springer.
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.
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.
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.
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).
Ibragimov, I. A., Linnik, Y. V. (1971). Independent and Stationary Sequences of Random Variables. Groningen: Wolters-Noordhoff.
Illian, J., Penttinen, A., Stoyan, H., Stoyan, D. (2008). Statistical Analysis and Modelling of Spatial Point Patterns. Chichester: Wiley.
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.
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).
Lindsay, B. G. (1988). Composite likelihood methods. Contemporary Mathematics, 80, 221–239.
Møller, J., Waagepetersen, R. P. (2003). Statistical Inference and Simulation for Spatial Point Processes. Boca Raton: Chapman& Hall/CRC.
Møller, J., Waagepetersen, R. P. (2007). Modern statistics for spatial point processes. Scandinavian Journal of Statistics, 34, 643–684.
Møller, J., Syversveen, A. R., Waagepetersen, R. P. (1998). Log Gaussian Cox processes. Scandinavian Journal of Statistics, 25, 451–482.
Nelder, J. A., Mead, R. (1965). A simplex method for function minimization. Computer Journal, 7, 308.
Ogata, Y., Katsura, K. (1991). Maximum likelihood estimates of fractal dimension for random spatial patterns. Biometrika, 78, 463–474.
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.
Rosenblatt, M. (1956). A central limit theorem and a strong mixing condition. Proceedings of the National Academy of Sciences, 42, 43–47.
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.
Stein, M. L. (1999). Interpolation of Spatial Data. New York: Springer.
Stoyan, D., Kendall, W. S., Mecke, J. (1995). Stochastic Geometry and its Applications (2nd ed.). Chichester: Wiley.
Tanaka, U., Ogata, Y., Stoyan, D. (2008). Parameter Estimation and Model Selection for Neyman–Scott Point Processes. Biometrical Journal, 50, 43–57.
Thomas, M. (1949). A generalization of Poisson’s binomial limit for use in ecology. Biometrika, 36, 18–25.
Waagepetersen, R. P. (2007). An estimating function approach to inference for inhomogeneous Neyman–Scott processes. Biometrics, 63, 252–258.
Wills, J. M. (1970). Zum Verhältnis von Volumen zu Oberfläche bei Convexen Körpern. Archiv der Mathematik, 21, 557–560.
Zhengyan, L., Chuanrong, L. (1996). Limit Theory for Mixing Dependent Random Variables. Dordrecht: Kluwer Academic Publishers.
Acknowledgments
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.
Author information
Authors and Affiliations
Corresponding author
About this article
Cite this article
Prokešová, M., Jensen, E.B.V. Asymptotic Palm likelihood theory for stationary point processes. Ann Inst Stat Math 65, 387–412 (2013). https://doi.org/10.1007/s10463-012-0376-7
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10463-012-0376-7