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Asymptotic Properties of the Approximate Inverse Estimator for Directional Distributions

Part of: Stochastic processes Geometric probability and stochastic geometry

Published online by Cambridge University Press:  04 January 2016

M. Riplinger  and
M. Spiess
M. Riplinger*
Affiliation:
Saarland University
M. Spiess*
Affiliation:
Ulm University
*
∗ Postal address: Institute of Applied Mathematics, Saarland University, 66041 Saarbrücken, Germany. Email address: riplinger@num.uni-sb.de
∗∗ Postal address: Institute of Stochastics, Ulm University, Helmholtzstr. 18, 89069 Ulm, Germany. Email address: malte.spiess@uni-ulm.de
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Abstract

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For stationary fiber processes, the estimation of the directional distribution is an important task. We consider a stereological approach, assuming that the intersection points of the process with a finite number of test hyperplanes can be observed in a bounded window. The intensity of these intersection processes is proportional to the cosine transform of the directional distribution. We use the approximate inverse method to invert the cosine transform and analyze asymptotic properties of the estimator in growing windows for Poisson line processes. We show almost-sure convergence of the estimator and derive Berry–Esseen bounds, including formulae for the variance.

Keywords

Cosine transformfiber processinverse problemPoisson processrose of intersectionsstereology

MSC classification

Primary: 60G10: Stationary processes
Secondary: 60D05: Geometric probability and stochastic geometry
Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 44 , Issue 4 , December 2012 , pp. 954 - 976
Copyright
© Applied Probability Trust 

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