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The Rao–Blackwell theorem in stereology and some counterexamples

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

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.

References

[1] Baddeley, A. J. (1991) Stereology. In Spatial Statistics and Digital Image Analysis, Chapter 10, pages 181–216. National Research Council USA, Washington DC, 1991.Google Scholar
[2] Baddeley, A. J. (1993) Stereology and survey sampling theory. Bull. Internat. Statist. Inst. 50(2), 435–449.Google Scholar
[3] Baddeley, A. J., Gundersen, H. J. G. and Cruz-Orive, L. M. (1986) Estimation of surface area from vertical sections. J. Microscopy 142, 259–276.Google Scholar
[4] Bahadur, R. R. (1954) Sufficiency and statistical decision functions. Ann. Math. Statist. 25, 423–462.Google Scholar
[5] Bellhouse, D. R. (1988) Systematic sampling. In Sampling, Volume 6 of Handbook of Statistics , ed. Krishnaiah, P. R. and Rao, C. R., Chapter 6. North-Holland, Amsterdam.Google Scholar
[6] Blackwell, D. (1947) Conditional expectation and unbiased sequential estimation. Ann. Math. Statist. 18, 105–110.Google Scholar
[7] Cox, D. R. and Hinkley, D. V. (1974) Theoretical Statistics. Chapman and Hall, London.Google Scholar
[8] Cressie, N. A. C. (1991) Statistics for Spatial Data. Wiley, New York.Google Scholar
[9] Cruz-Orive, L. M. (1980) Best linear unbiased estimators for stereology. Biometrics 36, 595–605.Google Scholar
[10] Cruz-Orive, L. M. (1982) The use of quadrats and test systems in stereology, including magnification corrections. J. Microscopy 125, 89–102.CrossRefGoogle Scholar
[11] Cruz-Orive, L. M. (1989) Second-order stereology: estimation of second moment volume measures. Acta Stereol. 8, 641–646.Google Scholar
[12] Cruz-Orive, L. M. and Roberts, N. (1933) Unbiased volume estimation with coaxial sections: an application to the human bladder. J. Microscopy 170, 25–33.Google Scholar
[13] Cruz-Orive, L. M. and Weibel, E. R. (1990) Recent stereological methods for cell biology: a brief survey. Amer. J. Physiol. 258, L148–L156. Lung Cell. Mol. Physiol. 2.Google Scholar
[14] Davy, P. J. (1978) Stereology: a statistical viewpoint. Ph.D. thesis, Australian National University.Google Scholar
[15] Davy, P. J. and Miles, R. E. (1977) Sampling theory for opaque spatial specimens. J. R. Statist. Soc. B 39, 56–65.Google Scholar
[16] Dhani, A. (1979) Problèmes d'éstimation en stéréologie. Mémoire, Licencié en sciences mathématiques, Faculté de Science, Facultés Universitaires de Notre Dame de la Paix, Namur, Belgium.Google Scholar
[17] Downie, A. S. (1991) Efficiency of statistics in stereology. Ph.D. thesis, Imperial College London.Google Scholar
[18] Grenander, U. (1950) Stochastic processes and statistical inference. Ark. Math. 1, 195–277.Google Scholar
[19] Gundersen, H. J. and Østerby, R. (1981) Optimizing sampling efficiency of stereological studies in biology: or ‘Do more less well!’. J. Microsc. 121, 65–74.CrossRefGoogle ScholarPubMed
[20] Gundersen, H. J. G., Boysen, M. and Reith, A. (1981). Comparison of semiautomatic digitizer-tablet and simple point counting performance in morphometry. Virchows Archiv B (Cell Pathol.) , 37, 317–325.CrossRefGoogle Scholar
[21] Gundersen, H. J. G., Boysen, M. and Reith, A. (1981) Digitizer-tablet or point counting in biomorphometry? In Proc. 3rd European Symp. Stereology , 3 (suppl.), 205–210.Google Scholar
[22] Gundersen, H. J. G. et al. (1988) Some new, simple and efficient stereological methods and their use in pathological research and diagnosis. Acta Pathol. Microbiol. Immunol. Scand. 96, 379–394.Google Scholar
[23] Halmos, P. and Savage, L. J. (1949) Application of the Radon–Nikodym theorem to the theory of sufficient statistics. Ann. Math. Statist. 20, 225–241.Google Scholar
[24] Jensen, E. B., Baddeley, A. J., Gundersen, H. J. G. and Sundberg, R. (1985) Recent trends in stereology. Internat. Statist. Rev. 53, 99–108.Google Scholar
[25] Jensen, E. B. and Gundersen, H. J. G. (1982) Stereological ratio estimation based on counts from integral test systems. J. Microscopy 125, 51–55.CrossRefGoogle Scholar
[26] Jensen, E. B. and Sundberg, R. (1986) Statistical models for stereological inference about spatial structures; on the applicability of best linear unbiased estimators in stereology. Biometrics 42, 735–751.Google Scholar
[27] Jensen, E. B. V. and Gundersen, H. J. G. (1993). The rotator. J. Microscopy 170, 35–44.Google Scholar
[28] Kendall, M. G. and Moran, P. A. P. (1963) Geometrical Probability. Griffin, London.Google Scholar
[29] Kieu, K. and Jensen, E. B. V. (1993) Stereological estimation based on isotropic slices through fixed points. J. Microscopy 170, 45–51.Google Scholar
[30] Lantuejoul, C. (1988) Some stereological and statistical consequences derived from Cartier's formula. J. Microscopy 151, 265–276.CrossRefGoogle Scholar
[31] Matern, B. (1986) Spatial Variation. Lecture Notes in Statistics 36. Springer-Verlag, New York.CrossRefGoogle Scholar
[32] Matern, B. (1989) Precision of area estimation: a numerical study. J. Microscopy 153, 269–284.Google Scholar
[33] Matheron, G. (1965) Les variables régionalisées et leur estimation . Masson, Paris.Google Scholar
[34] Matheron, G. (1973) The intrinsic random functions and their applications. Adv. Appl. Prob. 5, 439–468.Google Scholar
[35] Matheron, G. (1975) Random Sets and Integral Geometry . Wiley, New York.Google Scholar
[36] Mathieu, O., Cruz-Orive, L. M., Hoppeler, H. and Weibel, E. R. (1981) Measuring error and sampling variation in stereology: comparison of the efficiency of various methods for planar image analysis. J. Microscopy 121, 75–88.CrossRefGoogle ScholarPubMed
[37] Miles, R. E. (1978) The importance of proper model specification in stereology. In Geometrical Probability and Biological Structures: Buffon's 200th Anniversary , ed. Miles, R. E. and Serra, J., Lecture Notes in Biomathematics 23, pp. 115–136, Springer-Verlag, Berlin.Google Scholar
[38] Miles, R. E. (1978) The sampling, by quadrats, of planar aggregates. J. Microscopy 113, 257–267.Google Scholar
[39] Miles, R. E. and Davy, P. J. (1976) Precise and general conditions for the validity of a comprehensive set of stereological fundamental formulae. J. Microscopy 107, 211–226.CrossRefGoogle Scholar
[40] Miles, R. E. and Davy, P. J. (1977) On the choice of quadrats in stereology. J. Microscopy 110, 27–44.CrossRefGoogle Scholar
[41] Nagel, S. (1985) A best unbiased ratio-estimator in stereology. In Geobild' 85 , ed. Nagel, W., pp. 147–153, Friedrich-Schiller-Universität, Jena.Google Scholar
[49] Näther, W. (1985) Effective Observation of Random Fields. Teubner, Leipzig.Google Scholar
[43] Ohser, J. (1990) Grundlagen und praktische Möglichkeiten der Charakterisierung struktureller Inhomogenitäten von Werkstoffen. Dr. sc. techn. thesis, Bergakademie Freiberg, Freiberg (Sachsen), Germany.Google Scholar
[44] Ohser, J. (1991) Variances for different estimators of specific line length. Lecture at the meeting on Stochastic Geometry, Geometric Statistics, Stereology , Oberwolfach.Google Scholar
[45] Quenouille, M. H. (1949) Problems in plane sampling. Ann. Math. Statist. 20, 335–375.CrossRefGoogle Scholar
[46] Rao, C. R. (1945) Information and accuracy obtainable in estimation of statistical parameters. Bull. Calcutta Math. Soc. 37, 81–91.Google Scholar
[47] Ripley, B. D. (1981) Spatial Statistics. Wiley, New York.Google Scholar
[48] Santaló, L. A. (1976) Integral Geometry and Geometric Probability. Encyclopedia of Mathematics and its Applications , Vol. 1. Addison-Wesley, Reading, MA.Google Scholar
[49] Särndal, C. E. (1978) Design-based and model-based inference in survey sampling (with discussion). Scand. J. Statist. 5, 27–52.Google Scholar
[50] Schwandtke, A., Ohser, J. and Stoyan, D. (1987) Improved estimation in planar sampling. Acta Stereol. 6(2), 325–334.Google Scholar
[51] Smit, J. C. (1961) Estimation of the mean of a stationary stochastic process by equidistant observations. Trab. Estadist. 12, 35–45.CrossRefGoogle Scholar
[52] Stoyan, D. (1990). Stereology and stochastic geometry. Internat. Statist. Rev. 58, 227–242.Google Scholar
[53] Stoyan, D., Kendall, W. S. and Mecke, J. (1987) Stochastic Geometry and its Applications. Wiley, Chichester.Google Scholar
[54] Vilenkin, S. Ya. (1959) On the estimation of the mean in stationary processes. Theory Prob. Appl. 4, 415–416.Google Scholar
[55] Weibel, E. R. (1979) Stereological Methods, 1. Practical Methods for Biological Morphometry. Academic Press, London.Google Scholar
[56] Weibel, E. R. (1980) Stereological Methods, 2. Theoretical Foundations . Academic Press, London.Google Scholar
[57] Ylvisaker, D. (1975) Designs on random fields. In A Survey of Statistical Design and Linear Models , ed. Srivastava, J., pp. 593–607. North-Holland, Amsterdam.Google Scholar
[58] Zubrzycki, S. (1958) Remarks on random, stratified and systematic sampling in a plane. Colloq. Math. 6, 251–264.Google Scholar