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Estimation of the mean normal measure from flat sections

Published online by Cambridge University Press:  01 July 2016

Markus Kiderlen*
Affiliation:
University of Aarhus
*
Postal address: Department of Mathematical Sciences, University of Aarhus, Ny Munkegade Build. 1530, DK-8000 Aarhus C, Denmark. Email address: [email protected]
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Abstract

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We discuss the determination of the mean normal measure of a stationary random set Z ⊂ ℝd by taking measurements at the intersections of Z with k-dimensional planes. We show that mean normal measures of sections with vertical planes determine the mean normal measure of Z if k ≥ 3 or if k = 2 and an additional mild assumption holds. The mean normal measures of finitely many flat sections are not sufficient for this purpose. On the other hand, a discrete mean normal measure can be verified (i.e. an a priori guess can be confirmed or discarded) using mean normal measures of intersections with m suitably chosen planes when m ≥ ⌊d / k⌋ + 1. This even holds for almost all m-tuples of k-dimensional planes are viable for verification. A consistent estimator for the mean normal measure of Z, based on stereological measurements in vertical sections, is also presented.

Type
Stochastic Geometry and Statistical Applications
Copyright
Copyright © Applied Probability Trust 2008 

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