Abstract
There are three approaches for the estimation of the distribution function D(r) of distance to the nearest neighbour of a stationary point process: the border method, the Hanisch method and the Kaplan-Meier approach. The corresponding estimators and some modifications are compared with respect to bias and mean squared error (mse). Simulations for Poisson, cluster and hard-core processes show that the classical border estimator has good properties; still better is the Hanisch estimator. Typically, mse depends on r, having small values for small and large r and a maximum in between. The mse is not reduced if the exact intensity λ (if known) or intensity estimators from larger windows are built in the estimators of D(r); in contrast, the intensity estimator should have the same precision as that of λ D(r). In the case of replicated estimation from more than one window the best way of pooling the subwindow estimates is averaging by weights which are proportional to squared point numbers.
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References
Baddeley AJ, Moyeed RA, Howard CV, Boyde A (1993). Analysis of a three-dimensional point pattern with replication. Appl Statist 42:641–668
Baddeley AJ, Gill RD (1997). Kaplan-Meier estimators for interpoint distance distributions for spatial point processes. Ann Statist 25:263–292
Baddeley AJ (1999). Spatial sampling and censoring. In: Barndorff-Nielsen OE, Kendall WS, van Lieshout MNM (eds). Stochastic geometry Likelihood and computation. Chapman & Hall/CRC, Boca Raton and London, pp 37–78
Cressie N (1991). Statistics of spatial data. Wiley, New York
Diggle PJ (1979). On parameter estimation and goodness-of-fit testing for spatial point processes. Biometrics 35:87–101
Diggle PJ (1983). Statistical analysis of point processes. Chapman & Hall, London
Diggle PJ (2003). Statistical analysis of point processes, 2nd edn. Arnold, London
Diggle PJ, Lange N, Beneš FN (1991). Analysis of variance for replicated spatialo point patterns in clinical neuroanatomy. J Am Stat Assoc 86:618–625
Diggle PJ, Mateu J, Clough HE (2000). A comparison between parametric and non-parametric approaches to the analysis of replicated spatial point patterns. Adv Appl Prob 32:331–343
Hanisch K-H (1984). Some remarks on estimators of the distribution function of nearest neighbour distance in stationary spatial point patterns. Statistics 15:409–412
Møller J, Waagepetersen RP (2003). Statistical inference and simulation for spatial point processes. Chapman & Hall/CRC, Boca Raton, London
Ohser J, Mücklich F (2000). Statistical analysis of microstructures in materials science. Wiley, New York
Ripley BD (1977). Modelling spatial patterns. J R Stat Soc B39:172–212
Ripley BD (1988). Statistical inference for spatial processes. Cambridge University Press, Cambridge
Stoyan D, Kendall WS, Mecke J (1995). Stochastic geometry and its applications. Wiley, Chichester
Stoyan D, Stoyan H (2000). Improving ratio estimators of second order point proan characteristics. Scand J Statist 27:641–656
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Stoyan, D. On Estimators of the Nearest Neighbour Distance Distribution Function for Stationary Point Processes. Metrika 64, 139–150 (2006). https://doi.org/10.1007/s00184-006-0040-4
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DOI: https://doi.org/10.1007/s00184-006-0040-4
Keywords
- Stationary point process
- Nearest neighbour distance distribution
- Hanisch estimator
- Mean squared deviation
- Replication