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Estimation of the directional measure of planar random sets by digitization

Part of: General convexity Parametric inference Geometric probability and stochastic geometry

Published online by Cambridge University Press:  01 July 2016

Markus Kiderlen  and
Eva B. Vedel Jensen
Markus Kiderlen*
Affiliation:
University of Karlsruhe
Eva B. Vedel Jensen*
Affiliation:
University of Aarhus
*
∗ Postal address: Mathematical Institute II, University of Karlsruhe, D-76128 Karlsruhe, Germany. Email address: [email protected]
∗∗ Postal address: Department of Theoretical Statistics, Institute of Mathematics, University of Aarhus, NY Munkegade, DK-8000 Aarhus C, Denmark.
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Abstract

Estimation methods for the directional measure of a stationary planar random set Z, based only on discretized realizations of Z, are discussed. Properties of the discretized set that can be derived by comparing neighbouring grid points are used. Larger grid configurations of more than two grid points are considered. It is shown that the probabilities of observing the various types of configurations can be expressed in terms of the first contact distribution function of Z (with a finite structuring element). An important prerequisite result concerning deterministic dilation areas is also established. The inference on the mean normal measure based on 2×2 configurations is discussed in detail.

Keywords

Mean normal measureoriented rose of normal directionsdigitizationcontact distributionpoint configuration

MSC classification

Primary: 60D05: Geometric probability and stochastic geometry 62F10: Point estimation
Secondary: 52A10: Convex sets in $2$ dimensions (including convex curves)
Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 35 , Issue 3 , September 2003 , pp. 583 - 602
Copyright
Copyright © Applied Probability Trust 2003 

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