Advances in Applied Probability

Asymptotic properties of the approximate inverse estimator for directional distributions

M. Riplinger and M. Spiess

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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.

Article information

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

First available in Project Euclid: 5 December 2012

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

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


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.

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