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Variance prediction for pseudosystematic sampling on the sphere

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, Castellón
Luis M. Cruz-Orive*
Affiliation:
Universidad de Cantabria
*
∗ Postal address: Departament de Matemàtiques, Universitat Jaume I, Campus Riu Sec, E-12071 Castellón, Spain.
∗∗ Postal address: Departamento de Matemáticas, Estadística y Computación, Facultad de Ciencias, Universidad de Cantabria, Avenida Los Castros s/n, E-39005 Santander, Spain. Email address: lcruz@matesco.unican.es
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Abstract

Geometric sampling, and local stereology in particular, often require observations at isotropic random directions on the sphere, and some sort of systematic design on the sphere becomes necessary on grounds of efficiency and practical applicability. Typically, the relevant probes are of nucleator type, in which several rays may be contained in a sectioning plane through a fixed point (e.g. through a nucleolus within a biological cell). The latter requirement considerably reduces the choice of design in practice; in this paper, we concentrate on a nucleator design based on splitting the sphere into regions of equal area, but not of identical shape; this design is pseudosystematic rather than systematic in a strict sense. Firstly, we obtain useful exact representations of the variance of an estimator under pseudosystematic sampling on the sphere. Then we adopt a suitable covariogram model to obtain a variance predictor from a single sample of arbitrary size, and finally we examine the prediction accuracy by way of simulation on a synthetic particle model.

Keywords

Bernoulli polynomialcovariogramFourier analysisgeometric samplingnucleatorperiodic functionstereology

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 34 , Issue 3 , September 2002 , pp. 469 - 483
Copyright
Copyright © Applied Probability Trust 2002 

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Footnotes

Research supported by the Ministerio de Ciencia y Tecnologia (Spain) I+D project BSA2001-0803-C02.

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