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Statistics for Poisson Models of Overlapping Spheres

Part of: Global differential geometry Nonparametric inference Stochastic processes General convexity Geometric probability and stochastic geometry Normed linear spaces and Banach spaces; Banach lattices

Published online by Cambridge University Press:  22 February 2016

Daniel Hug ,
Günter Last ,
Zbyněk Pawlas  and
Wolfgang Weil
Daniel Hug*
Affiliation:
Karlsruhe Institute of Technology
Günter Last*
Affiliation:
Karlsruhe Institute of Technology
Zbyněk Pawlas*
Affiliation:
Charles University in Prague
Wolfgang Weil*
Affiliation:
Karlsruhe Institute of Technology
*
∗ Postal address: Department of Mathematics, Karlsruhe Institute of Technology, D-76128 Karlsruhe, Germany.
∗ Postal address: Department of Mathematics, Karlsruhe Institute of Technology, D-76128 Karlsruhe, Germany.
∗∗∗∗ Postal address: Department of Probability and Mathematical Statistics, Faculty of Mathematics and Physics, Charles University, Sokolovská 83, 18675 Praha 8, Czech Republic. Email address: zbynek.pawlas@mff.cuni.cz
∗ Postal address: Department of Mathematics, Karlsruhe Institute of Technology, D-76128 Karlsruhe, Germany.
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Abstract

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In this paper we consider the stationary Poisson Boolean model with spherical grains and propose a family of nonparametric estimators for the radius distribution. These estimators are based on observed distances and radii, weighted in an appropriate way. They are ratio unbiased and asymptotically consistent for a growing observation window. We show that the asymptotic variance exists and is given by a fairly explicit integral expression. Asymptotic normality is established under a suitable integrability assumption on the weight function. We also provide a short discussion of related estimators as well as a simulation study.

Keywords

Stochastic geometryspatial statisticcontact distribution functionBoolean modelspherical typical grainpoint processnonparametric estimationradius distributionasymptotic normality

MSC classification

Primary: 60D05: Geometric probability and stochastic geometry 60G57: Random measures 52A21: Finite-dimensional Banach spaces (including special norms, zonoids, etc.)
Secondary: 60G55: Point processes 52A22: Random convex sets and integral geometry 52A20: Convex sets in $n$ dimensions (including convex hypersurfaces) 53C65: Integral geometry 46B20: Geometry and structure of normed linear spaces 62G05: Estimation
Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 46 , Issue 4 , December 2014 , pp. 937 - 962
Copyright
© Applied Probability Trust 

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