Infill asymptotics for adaptive kernel estimators of spatial intensity. (English) Zbl 1521.62185
Summary: We apply the Abramson principle to define adaptive kernel estimators for the intensity function of a spatial point process. We derive asymptotic expansions for the bias and variance under the regime that \(n\) independent copies of a simple point process in Euclidean space are superposed. The method is illustrated by means of a simple example and applied to tornado data.
© 2021 John Wiley & Sons Australia, Ltd
© 2021 John Wiley & Sons Australia, Ltd
MSC:
| 62M30 | Inference from spatial processes |
| 60G55 | Point processes (e.g., Poisson, Cox, Hawkes processes) |
| 62G07 | Density estimation |
Keywords:
adaptive kernel estimator; bandwidth; infill asymptotics; intensity function; mean squared error; point processReferences:
| [1] | Abramson, I.A. (1982). On bandwidth variation in kernel estimates - A square root law. Annals of Statistics10, 1217-1223. · Zbl 0507.62040 |
| [2] | Baddeley, A., Rubak, E. & Turner, R.(2016). Spatial Point Patterns: Methodology and Applications with R. Boca Raton: CRC Press. · Zbl 1357.62001 |
| [3] | Berman, M. & Diggle, P. (1989). Estimating weighted integrals of the second‐order intensity of a spatial point process. Journal of the Royal Statistical Society, series B51, 81-92. · Zbl 0671.62043 |
| [4] | Bowman, A.W. & Azzalini, A. (1997). Applied Smoothing Techniques for Data Analysis. The Kernel Approach with S‐Plus Illustrations. Oxford: University Press. · Zbl 0889.62027 |
| [5] | Bowman, A.W. & Foster, P.J. (1993). Adaptive smoothing and density‐based tests of multivariate normality. Journal of the American Statistical Association88, 529-537. · Zbl 0775.62086 |
| [6] | Brooks, M.M. & Marron, J.S. (1991). Asymptotic optimality of the least‐squares cross‐validation bandwidth for kernel estimates of intensity functions. Stochastic Processes and their Applications38, 157-165. · Zbl 0724.62084 |
| [7] | Chacón, J.E. & Duong, T. (2018). Multivariate Kernel Smoothing and its Applications. Boca Raton: CRC Press. · Zbl 1402.62003 |
| [8] | Chiu, S.N., Stoyan, D., Kendall, W.S. & Mecke, J. (2013). Stochastic Geometry and its Applications, 3rd edn. Chichester: Wiley. · Zbl 1291.60005 |
| [9] | Coles, P. & Jones, B. (1991). A lognormal model for the cosmological mass distribution. Monthly Notices of the Royal Astronomical Society248, 1-13. |
| [10] | Cronie, O. & van Lieshout, M.N.M. (2018). A non‐model based approach to bandwidth selection for kernel estimators of spatial intensity functions. Biometrika105, 455-462. · Zbl 07072424 |
| [11] | Davies, T.M. & Baddeley, A. (2018). Fast computation of spatially adaptive kernel estimates. Statistics and Computing28, 937-956. · Zbl 1384.62175 |
| [12] | Davies, T.M., Flynn, C.R. & Hazelton, M.L. (2018). On the utility of asymptotic bandwidth selectors for spatially adaptive kernel density estimation. Statistics and Probability Letters138, 75-81. · Zbl 1463.62043 |
| [13] | Diggle, P.J. (1985). A kernel method for smoothing point process data. Journal of Applied Statistics34, 138-147. · Zbl 0584.62140 |
| [14] | Duong, T. (2015). Spherically symmetric multivariate Beta family kernels. Statistics and Probability Letters104, 141-145. · Zbl 1396.62100 |
| [15] | EngelJ., Herrmann, E. & Gasser, T. (1994). An iterative bandwidth selector for kernel estimation of densities and their derivatives. Journal of Nonparametric Statistics4, 21-34. · Zbl 1380.62146 |
| [16] | Hall, P., Hu, T.C. & Marron, J.S. (1995). Improved variable window kernel estimates of probability densities. Annals of Statistics23, 1-10. · Zbl 0822.62026 |
| [17] | Illian, J., Penttinen, A., Stoyan, H. & Stoyan, D.. (2008). Statistical Analysis and Modelling of Spatial Point Patterns. Chichester: Wiley. · Zbl 1197.62135 |
| [18] | van Lieshout, M.N.M. (2012). On estimation of the intensity function of a point process. Methodology and Computing in Applied Probability14, 567-578. · Zbl 1274.60157 |
| [19] | van Lieshout, M.N.M. (2019). Theory of Spatial Statistics: A Concise Introduction. Boca Raton: CRC Press. · Zbl 1462.62003 |
| [20] | van Lieshout, M.N.M. (2020). Infill asymptotics and bandwidth selection for kernel estimators of spatial intensity functions. Methodology and Computing in Applied Probability22, 995-1008. · Zbl 1460.60042 |
| [21] | Lo, P.H. (2017). An Iterative Plug‐in Algorithm for Optimal Bandwidth Selection in Kernel Intensity Estimation for Spatial Data. Technical University of Kaiserslautern: PhD Thesis. |
| [22] | Loader, C. (1999). Local Regression and Likelihood. New York: Springer. · Zbl 0929.62046 |
| [23] | M⊘ller, J. & SyversveenA.R. & and Waagepetersen, R.P. (1998). Log Gaussian Cox processes. Scandinavian Journal of Statistics25, 451-482. · Zbl 0931.60038 |
| [24] | Ripley, B.D. (1988). Statistical Inference for Spatial Processes. Cambridge: University Press. · Zbl 0716.62100 |
| [25] | Scott, D.W. (1992). Multivariate Density Estimation: Theory, Practice and Visualization. New York: Wiley. · Zbl 0850.62006 |
| [26] | Silverman, B.W. (1986). Density Estimation for Statistics and Data Analysis. Boca Raton: Chapman & Hall. · Zbl 0617.62042 |
| [27] | Wand, M.P. & Jones, M.C. (1995). Kernel Smoothing. Boca Raton: Chapman & Hall. · Zbl 0854.62043 |
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