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Infill Asymptotics and Bandwidth Selection for Kernel Estimators of Spatial Intensity Functions

  • M. N. M. van LieshoutEmail author
Open Access
Article
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Abstract

We investigate the asymptotic mean squared error of kernel estimators of the intensity function of a spatial point process. We derive expansions for the bias and variance in the scenario that n independent copies of a point process in \(\mathbb {R}^{d}\) are superposed. When the same bandwidth is used in all d dimensions, we show that an optimal bandwidth exists and is of the order n− 1/(d+ 4) under appropriate smoothness conditions on the true intensity function.

Keywords

Bandwidth Infill asymptotics Intensity function Kernel estimator Mean squared error Point process 

Mathematics Subject Classification (2010)

60G55 62G07 60D05 

Notes

Acknowledgements

We are grateful to the referee and associate editor for their careful reading of the manuscript.

References

  1. Abramson IA (1982) On bandwidth variation in kernel estimates – a square root law. Ann Statist 10:1217–1223MathSciNetCrossRefGoogle Scholar
  2. Berman M, Diggle PJ (1989) Estimating weighted integrals of the second-order intensity of a spatial point process. J R Stat Soc Ser B 51:81–92MathSciNetzbMATHGoogle Scholar
  3. Bowman AW, Azzalini A (1997) Applied smoothing techniques for data analysis. The kernel approach with S-Plus illustrations. University Press, OxfordzbMATHGoogle Scholar
  4. Brooks MM, Marron JS (1991) Asymptotic optimality of the least-squares cross-validation bandwidth for kernel estimates of intensity functions. Stochastic Process Appl 38:157–165MathSciNetCrossRefGoogle Scholar
  5. Chiu SN, Stoyan D, Kendall WS, Mecke J (2013) Stochastic geometry and its applications, 3rd edn. Wiley, ChichesterCrossRefGoogle Scholar
  6. Cowling A, Hall P, Phillips MJ (1996) Bootstrap confidence regions for the intensity of a Poisson point process. J Amer Statist Assoc 91:1516–1524MathSciNetCrossRefGoogle Scholar
  7. Cronie O, Van Lieshout MNM (2018) A non-model based approach to bandwidth selection for kernel estimators of spatial intensity functions. Biometrika 105:455–462MathSciNetCrossRefGoogle Scholar
  8. Diggle PJ (1985) A kernel method for smoothing point process data. J Appl Stat 34:138–147CrossRefGoogle Scholar
  9. Engel J, Herrmann E, Gasser T (1994) An iterative bandwidth selector for kernel estimation of densities and their derivatives. J Nonparametr Statist 4:21–34MathSciNetCrossRefGoogle Scholar
  10. Fuentes–Santos I, González–Manteiga W, Mateu J (2016) Consistent smooth bootstrap kernel intensity estimation for inhomogeneous spatial Poisson point processes. Scand J Stat 43:416–435MathSciNetCrossRefGoogle Scholar
  11. Granville V (1998) Estimation of the intensity of a Poisson point process by means of nearest neighbour distances. Stat Neerl 52:112–124MathSciNetCrossRefGoogle Scholar
  12. Lo PH (2017) An iterative plug-in algorithm for optimal bandwidth selection in kernel intensity estimation for spatial data. PhD thesis, Technical University of KaiserslauternGoogle Scholar
  13. Ord JK (1978) How many trees in a forest? Math Sci 3:23–33Google Scholar
  14. Parzen E (1962) On estimation of a probability density function and mode. Ann Math Statist 33:1065–1076MathSciNetCrossRefGoogle Scholar
  15. Ripley BD (1988) Statistical inference for spatial processes. University Press, CambridgeCrossRefGoogle Scholar
  16. Schaap WE (2007) DTFE. The Delaunay tessellation field estimator. PhD Thesis, University of GroningenGoogle Scholar
  17. Schaap WE, Van de Weygaert R (2000) Letter to the editor. Continuous fields and discrete samples: reconstruction through Delaunay tessellations. Astronom Astrophys 363:L29–L32Google Scholar
  18. Silverman BW (1986) Density estimation for statistics and data analysis. Chapman & Hall, Boca RatonCrossRefGoogle Scholar
  19. Van Lieshout MNM (2012) On estimation of the intensity function of a point process. Methodol Comput Appl Probab 14:567–578MathSciNetCrossRefGoogle Scholar
  20. Wand MP, Jones MC (1994) Kernel smoothing. Chapman & Hall, Boca RatonCrossRefGoogle Scholar

Copyright information

© The Author(s) 2019

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.CWIAmsterdamThe Netherlands
  2. 2.University of TwenteEnschedeThe Netherlands

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