Advances in Applied Probability

Statistics for Poisson models of overlapping spheres

Daniel Hug, Günter Last, Zbynȫk Pawlas, and Wolfgang Weil

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Abstract

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.

Article information

Source
Adv. in Appl. Probab., Volume 46, Number 4 (2014), 937-962.

Dates
First available in Project Euclid: 12 December 2014

Permanent link to this document
https://projecteuclid.org/euclid.aap/1418396238

Digital Object Identifier
doi:10.1239/aap/1418396238

Mathematical Reviews number (MathSciNet)
MR3290424

Zentralblatt MATH identifier
1319.60014

Subjects
Primary: 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65] 60G57: Random measures 52A21: Finite-dimensional Banach spaces (including special norms, zonoids, etc.) [See also 46Bxx]
Secondary: 60G55: Point processes 52A22: Random convex sets and integral geometry [See also 53C65, 60D05] 52A20: Convex sets in n dimensions (including convex hypersurfaces) [See also 53A07, 53C45] 53C65: Integral geometry [See also 52A22, 60D05]; differential forms, currents, etc. [See mainly 58Axx] 46B20: Geometry and structure of normed linear spaces 62G05: Estimation

Keywords
Stochastic geometry spatial statistic contact distribution function Boolean model spherical typical grain point process nonparametric estimation radius distribution asymptotic normality

Citation

Hug, Daniel; Last, Günter; Pawlas, Zbynȫk; Weil, Wolfgang. Statistics for Poisson models of overlapping spheres. Adv. in Appl. Probab. 46 (2014), no. 4, 937--962. doi:10.1239/aap/1418396238. https://projecteuclid.org/euclid.aap/1418396238


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