×

Infill asymptotics and bandwidth selection for kernel estimators of spatial intensity functions. (English) Zbl 1460.60042

Summary: 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.

MSC:

60G55 Point processes (e.g., Poisson, Cox, Hawkes processes)
62G07 Density estimation
60D05 Geometric probability and stochastic geometry

Software:

KernSmooth; spatial; DTFE
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Abramson, IA, On bandwidth variation in kernel estimates – a square root law, Ann Statist, 10, 1217-1223 (1982) · Zbl 0507.62040
[2] Berman, M.; Diggle, PJ, Estimating weighted integrals of the second-order intensity of a spatial point process, J R Stat Soc Ser B, 51, 81-92 (1989) · Zbl 0671.62043
[3] Bowman, AW; Azzalini, A., Applied smoothing techniques for data analysis. The kernel approach with S-Plus illustrations (1997), Oxford: University Press, Oxford · Zbl 0889.62027
[4] Brooks, MM; Marron, JS, Asymptotic optimality of the least-squares cross-validation bandwidth for kernel estimates of intensity functions, Stochastic Process Appl, 38, 157-165 (1991) · Zbl 0724.62084
[5] Chiu, SN; Stoyan, D.; Kendall, WS; Mecke, J., Stochastic geometry and its applications (2013), Chichester: Wiley, Chichester
[6] Cowling, A.; Hall, P.; Phillips, MJ, Bootstrap confidence regions for the intensity of a Poisson point process, J Amer Statist Assoc, 91, 1516-1524 (1996) · Zbl 0882.62078
[7] Cronie, O.; Van Lieshout, MNM, A non-model based approach to bandwidth selection for kernel estimators of spatial intensity functions, Biometrika, 105, 455-462 (2018) · Zbl 07072424
[8] Diggle, PJ, A kernel method for smoothing point process data, J Appl Stat, 34, 138-147 (1985) · Zbl 0584.62140
[9] Engel, J.; Herrmann, E.; Gasser, T., An iterative bandwidth selector for kernel estimation of densities and their derivatives, J Nonparametr Statist, 4, 21-34 (1994) · Zbl 1380.62146
[10] Fuentes-Santos, I.; González-Manteiga, W.; Mateu, J., Consistent smooth bootstrap kernel intensity estimation for inhomogeneous spatial Poisson point processes, Scand J Stat, 43, 416-435 (2016) · Zbl 1382.60073
[11] Granville, V., Estimation of the intensity of a Poisson point process by means of nearest neighbour distances, Stat Neerl, 52, 112-124 (1998) · Zbl 0937.62085
[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 Kaiserslautern
[13] Ord, JK, How many trees in a forest?, Math Sci, 3, 23-33 (1978)
[14] Parzen, E., On estimation of a probability density function and mode, Ann Math Statist, 33, 1065-1076 (1962) · Zbl 0116.11302
[15] Ripley, BD, Statistical inference for spatial processes (1988), Cambridge: University Press, Cambridge
[16] Schaap WE (2007) DTFE. The Delaunay tessellation field estimator. PhD Thesis, University of Groningen
[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-L32
[18] Silverman, BW, Density estimation for statistics and data analysis (1986), Boca Raton: Chapman & Hall, Boca Raton
[19] Van Lieshout, MNM, On estimation of the intensity function of a point process, Methodol Comput Appl Probab, 14, 567-578 (2012) · Zbl 1274.60157
[20] Wand, MP; Jones, MC, Kernel smoothing (1994), Boca Raton: Chapman & Hall, Boca Raton
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.