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Variance prediction under systematic sampling with geometric probes

Part of: Global differential geometry General convexity Geometric probability and stochastic geometry

Published online by Cambridge University Press:  01 July 2016

Ximo Gual Arnau  and
Luis M. Cruz-Orive
Ximo Gual Arnau*
Affiliation:
Universitat Jaume I
Luis M. Cruz-Orive*
Affiliation:
University of Cantabria
*
∗ Postal address: Departament de Matematiques, Universitat Jaume I, Campus Riu Sec, E-12071 Castellón, Spain.
∗∗ Postal address: Departamento de Matemáticas, Estadística y Computación, Facultad de Ciencias, Universidad de Cantabria, Avda. Los Castros s/n, E-39005 Santander, Spain. Email address: [email protected]
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Abstract

In design stereology, and in the context of geometric sampling in general, the problem often arises of estimating the integral of a bounded non-random function over a bounded manifold D ⊂ ℝn by systematic sampling with geometric probes. Variance predictors, often based on Matheron's theory of regionalized variables, are available when the relevant function is sampled at the points of a grid intersecting D, but not when the dimension of the probes is greater than zero. For instance, the volume of a bounded object may be estimated using parallel systematic planes, which amounts to sampling on ℝ1 with systematic points, or using parallel systematic slabs of thickness t > 0, which amounts to sampling on ℝ1 with non-overlapping systematic segments of length t > 0. Useful variance predictors exist for the former case, but not for the latter. In this paper we set out a general scheme to predict estimation variances when the dimension of either D, or of the probes, is n. We make some progress when both dimensions are equal to n, and obtain explicit results for n = 1 (e.g. for systematic slice sampling). We check and illustrate our results for the volume estimators of ellipsoids and of rat lung.

Keywords

Cavalieri estimatorcovariogramellipsoidEuler-MacLaurin summation formulaFourier transformregularizationstereologysystematic samplingtest systemvariance prediction

MSC classification

Primary: 60D05: Geometric probability and stochastic geometry
Secondary: 52A22: Random convex sets and integral geometry 53C65: Integral geometry
Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 30 , Issue 4 , December 1998 , pp. 889 - 903
Copyright
Copyright © Applied Probability Trust 1998 

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Footnotes

Research supported by the Fundació Caixa Castelló grant no. P1A-94-24 and the Dirección General de Investigación Cientifica y Técnica Project No. #PB94-1058.

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