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Systematic sampling on the circle and on the sphere

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

Published online by Cambridge University Press:  19 February 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. Email address: gual@mat.uji.es
∗∗ 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: lcruz@matesco.unican.es
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Abstract

Useful approximations have been developed along the years to predict the precision of systematic sampling for measurable functions of a bounded support in ℝd. Recently, the theory of systematic sampling on ℝ has received a thrust. In geometric sampling, design based unbiased estimators exist, however, which imply systematic sampling on the circle (𝕊1) and the semicircle (ℍ1); the planimeter estimator of an area, or the Buffon-Steinhaus estimator of curve length in the plane constitute popular examples. Over the last two decades, many other estimators of geometric measures have been obtained which imply systematic sampling also on the sphere (𝕊2). In this paper we adapt the theory available for non-periodic functions of bounded support on ℝ to periodic functions, and thereby to 𝕊1 and ℍ1, and we obtain new estimators of the corresponding variance approximations. Further we consider - we believe for the first time - the problem of predicting the precision of systematic sampling in 𝕊2. The paper starts with a historical perspective, and ends with suggestions for further research.

Keywords

CovariogramEuler-MacLaurin summation formulaFourier seriesgeometric samplingnucleatorspherical harmonicsstereologytransitive theory

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 32 , Issue 3 , September 2000 , pp. 628 - 647
Copyright
Copyright © Applied Probability Trust 2000 

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Footnotes

Research supported by the Fundaci'o Caixa Castell' grant no. P1A-94-24 and the Direcci-on General de Ensenanza Superior (DGES) grant no. PM97-0043.

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