Empirical Mark Covariance and Product Density Function of Stationary Marked Point Processes—A Survey on Asymptotic Results

Lothar Heinrich (), Stella Klein () and Martin Moser ()
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Lothar Heinrich: Augsburg University
Stella Klein: Augsburg University
Martin Moser: Munich University of Technology

Methodology and Computing in Applied Probability, 2014, vol. 16, issue 2, 283-293

Abstract: Abstract Marked point processes are stochastic models to describe random patterns of marked points {[X i ,M i ], i ≥ 1} in some bounded subset of the d-dimensional Euclidean space (usually d = 1, 2 or 3 in applications), where each point X i carries additional random information expressed as mark M i taking values in some metric space. To study the correlations between distinct points and between marks located at distinct points we use kernel-type estimators of the second-order product density and the mark covariance function of a spatially homogeneous marked point process. Both functions and their empirical counterparts are suitable characteristics to identify point process models by construction of statistical goodness-of-fit tests.

Keywords: Typical mark; Kernel estimation; Empirical mark covariance function; Integrated squared error; Weak consistency; Central limit theorem; Mark cumulant measure; Brillinger-mixing marked point processes; Primary 60G55, 62F12; Secondary 60F05, 60G60 (search for similar items in EconPapers)
Date: 2014
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DOI: 10.1007/s11009-012-9314-7

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