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The Rao–Blackwell theorem in stereology and some counterexamples

Part of: Geometric probability and stochastic geometry Sampling theory, sample surveys Inference from stochastic processes Sufficiency and information

Published online by Cambridge University Press:  01 July 2016

A. J. Baddeley  and
L. M. Cruz-Orive
A. J. Baddeley*
Affiliation:
University of Western Australia
L. M. Cruz-Orive*
Affiliation:
Universität Bern
*
* Postal address: Department of Mathematics, University of Western Australia, Nedlands, WA 6009, Australia.
** Present address: Departamento de Matemáticas, Estadística y Computación, Facultad de Ciencias, Universidad de Cantabria, Avda Los Castros s/n, E-39005 Santander, Spain.
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Abstract

A version of the Rao–Blackwell theorem is shown to apply to most, but not all, stereological sampling designs. Estimators based on random test grids typically have larger variance than quadrat estimators; random s-dimensional samples are worse than random r-dimensional samples for s < r. Furthermore, the standard stereological ratio estimators of different dimensions are canonically related to each other by the Rao–Blackwell process. However, there are realistic cases where sampling with a lower-dimensional probe increases efficiency. For example, estimators based on (conditionally) non-randomised test point grids may have smaller variance than quadrat estimators. Relative efficiency is related to issues in geostatistics and the theory of wide-sense stationary random fields. A uniform minimum variance unbiased estimator typically does not exist in our context.

Keywords

CARTIER'S FORMULACOMPLETE σ-ALGEBRACOVARIANCE FUNCTIONEFFICIENT ESTIMATIONGEOSTATISTICSGRENANDER'S THEOREMPOINT COUNTINGQUADRAT SAMPLINGRANDOM CLOSED SETSRANDOMISED SUBSAMPLINGSMIT'S PARADOXSPATIAL SAMPLING DESIGNSPATIAL STATISTICSSTOCHASTIC GEOMETRYSUFFICIENT (σ-ALGEBRATEST SYSTEMSWIDE-SENSE STATIONARY RANDOM FIELDS

MSC classification

Primary: 60D05: Geometric probability and stochastic geometry
Secondary: 62D05: Sampling theory, sample surveys 62M30: Spatial processes 62B05: Sufficient statistics and fields
Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 27 , Issue 1 , March 1995 , pp. 2 - 19
Copyright
Copyright © Applied Probability Trust 1995 

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Footnotes

The original version of this paper was presented at the International Workshop on Stochastic Geometry, Stereology and Image Analysis held at the Universidad Internacional Menendez Pelayo, Valencia, Spain, on 21–24 September 1993.

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