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Estimation of the intensity of stationary flat processes

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

Published online by Cambridge University Press:  01 July 2016

Katja Schladitz
Katja Schladitz*
Affiliation:
ITWM Kaiserslautern
*
∗ Postal address: ITWM (Institut für Techno- und Wirtschaftsmathematik), Erwin-Schrödinger-Straße, 67663 Kaiserslautern, Germany. Email address: schlad@itwm.uni-kl.de
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Abstract

The intensity of a stationary process of k-dimensional affine subspaces (k-flats) of ℝd with directional distribution from a given family R is estimated by observing the process in a compact window. To this end we introduce a type of unbiased estimator (the R-estimator) using the available information about the directional distribution.

Special cases are estimators for the intensity of stationary k-flat processes (1) with known directional distribution, (2) with directional distribution invariant with respect to a subgroup of the group of rotations in ℝd and (3) with unknown directional distribution.

We give sufficient conditions for the R-estimator to be the uniformly best unbiased estimator for the intensity of stationary Poisson k-flat processes with directional distribution in R. Equivalent statements for certain types of stationary Cox flat processes can be deduced directly from the results in the Poisson case.

Moreover, we consider stationary ergodic flat processes with directional distribution in R and general stationary flat processes with unknown directional distribution, all with a non-degeneracy property. In both cases our estimator turns out to be the uniformly best unbiased estimator from a restricted set of estimators. The result for general stationary flat processes is proved with the help of a factorization result for the second factorial moment measure.

Keywords

Stationary hyperplane processstationary line processPoisson flat processesintensityunbiased estimatorDavidson's conjecturesecond factorial moment measuresufficiencysymmetric completeness

MSC classification

Primary: 60D05: Geometric probability and stochastic geometry
Secondary: 52A22: Random convex sets and integral geometry 62G05: Estimation
Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 32 , Issue 1 , March 2000 , pp. 114 - 139
Copyright
Copyright © Applied Probability Trust 2000 

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