Advances in Applied Probability

Asymptotic properties of the approximate inverse estimator for directional distributions

M. Riplinger and M. Spiess

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Abstract

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.

Article information

Source
Adv. in Appl. Probab., Volume 44, Number 4 (2012), 954-976.

Dates
First available in Project Euclid: 5 December 2012

Permanent link to this document
https://projecteuclid.org/euclid.aap/1354716585

Digital Object Identifier
doi:10.1239/aap/1354716585

Mathematical Reviews number (MathSciNet)
MR3052845

Zentralblatt MATH identifier
1275.62043

Subjects
Primary: 60G10: Stationary processes
Secondary: 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65]

Keywords
Cosine transform fiber process inverse problem Poisson process rose of intersections stereology

Citation

Riplinger, M.; Spiess, M. Asymptotic properties of the approximate inverse estimator for directional distributions. Adv. in Appl. Probab. 44 (2012), no. 4, 954--976. doi:10.1239/aap/1354716585. https://projecteuclid.org/euclid.aap/1354716585


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