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

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  • M. N. M. Lieshout

    (CWI
    University of Twente)

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 ℝ d $\mathbb {R}^{d}$ 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.

Suggested Citation

  • M. N. M. Lieshout, 2020. "Infill Asymptotics and Bandwidth Selection for Kernel Estimators of Spatial Intensity Functions," Methodology and Computing in Applied Probability, Springer, vol. 22(3), pages 995-1008, September.
    • Handle: RePEc:spr:metcap:v:22:y:2020:i:3:d:10.1007_s11009-019-09749-x
    DOI: 10.1007/s11009-019-09749-x
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    References listed on IDEAS

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    1. V. Granville, 1998. "Estimation of the intensity of a Poisson point process by means of nearest neighbor distances," Statistica Neerlandica, Netherlands Society for Statistics and Operations Research, vol. 52(1), pages 112-124, March.
    2. O Cronie & M N M Van Lieshout, 2018. "A non-model-based approach to bandwidth selection for kernel estimators of spatial intensity functions," Biometrika, Biometrika Trust, vol. 105(2), pages 455-462.
    3. Marie-Colette N. M. Lieshout, 2012. "On Estimation of the Intensity Function of a Point Process," Methodology and Computing in Applied Probability, Springer, vol. 14(3), pages 567-578, September.
    4. Isabel Fuentes-Santos & Wenceslao González-Manteiga & Jorge Mateu, 2016. "Consistent Smooth Bootstrap Kernel Intensity Estimation for Inhomogeneous Spatial Poisson Point Processes," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 43(2), pages 416-435, June.
    5. Peter Diggle, 1985. "A Kernel Method for Smoothing Point Process Data," Journal of the Royal Statistical Society Series C, Royal Statistical Society, vol. 34(2), pages 138-147, June.
    6. Brooks, Maria Mori & Marron, J. Stephen, 1991. "Asymptotic optimality of the least-squares cross-validation bandwidth for kernel estimates of intensity functions," Stochastic Processes and their Applications, Elsevier, vol. 38(1), pages 157-165, June.
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