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Second-order smoothing of spatial point patterns with small sample sizes: a family of kernels

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Abstract

We consider kernel-based non-parametric estimation of second-order product densities of spatial point patterns. We present a new family of optimal and positive kernels that shows less variance and more flexibility than optimal kernels. This family generalises most of the classical and widely used kernel functions, such as Box or Epanechnikov kernels. We propose an alternative asymptotically unbiased estimator for the second-order product density function, and compare the performance of the estimator for several members of the family of optimal and positive kernels through MISE and relative efficiency. We present a simulation study to analyse the behaviour of such kernel functions, for three different spatial structures, for which we know the exact analytical form of the product density, and under small sample sizes. Some known datasets are revisited, and we also analyse the IMD dataset in the Rhineland Regional Council in Germany.

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Acknowledgments

The research work of Francisco J. Rodríguez-Cortés’s was supported by Grant P1-1B2012-52 and Jorge Mateu’s research was supported by Grant MTM2010-14961 from Ministry of Education. We also would like to thank Jonatan A. González for their valuable advice. We sincerely thank two referees and the Associate Editor for helpful comments and suggestions which led to improve the paper.

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  1. Department of Mathematics, University Jaume I, Campus Riu Sec, 12071, Castellón, Spain

    F. J. Rodríguez-Cortés & J. Mateu

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  1. F. J. Rodríguez-Cortés
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  2. J. Mateu
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Correspondence to F. J. Rodríguez-Cortés.

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Rodríguez-Cortés, F.J., Mateu, J. Second-order smoothing of spatial point patterns with small sample sizes: a family of kernels. Stoch Environ Res Risk Assess 29, 295–308 (2015). https://doi.org/10.1007/s00477-014-0944-x

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