Hostname: page-component-848d4c4894-v5vhk Total loading time: 0 Render date: 2024-07-04T07:31:34.323Z Has data issue: false hasContentIssue false
  • English
  • Français

Consistency in systematic sampling

Part of: Global differential geometry Geometric probability and stochastic geometry

Published online by Cambridge University Press:  01 July 2016

X. Gual Arnau  and
L. M. Cruz-Orive
X. Gual Arnau*
Affiliation:
Universitat Jaume I
L. M. Cruz-Orive*
Affiliation:
Universidad de Cantabria and Universität Bern
*
∗ Postal address: Departament de Matemàtiques, Penyeta Roja, Universitat Jaume I, E-12071 Castellón, Spain.
∗∗ Postal address: Departamento de Matemáticas, Estadística y Computación, Universidad de Cantabria, Avda. Los Castros, E-39005 Santander, Spain.
Get access Rights & Permissions [Opens in a new window]

Abstract

In design-based stereology, fixed parameters (such as volume, surface area, curve length, feature number, connectivity) of a non-random geometrical object are estimated by intersecting the object with randomly located and oriented geometrical probes (e.g. test slabs, planes, lines, points). Estimation accuracy may in principle be increased by increasing the number of probes, which are usually laid in a systematic pattern. An important prerequisite to increase accuracy, however, is that the relevant estimators are unbiased and consistent. The purpose of this paper is therefore to give sufficient conditions for the unbiasedness and strong consistency of design-based stereological estimators obtained by systematic sampling. Relevant mechanisms to increase sample size, compatible with stereological practice, are considered.

Keywords

CONSISTENCYDESIGN BASED INFERENCELATTICE RETRACTMODEL BASED INFERENCESTEREOLOGYSYSTEMATIC SAMPLINGTEST SYSTEMUNBIASEDNESS

MSC classification

Primary: 60D05: Geometric probability and stochastic geometry
Secondary: 53C65: Integral geometry
Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 28 , Issue 4 , December 1996 , pp. 982 - 992
Copyright
Copyright © Applied Probability Trust 1996 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

Work supported by the Swiss National Science Foundation Grant #31-28610.90, Dirección General de Investigación Científica y Técnica Grant #PB94-1058 and Fundació Caixa Castelló Grant #P1A-94-24.

References

[1] Baddeley, A. J. (1991) Stereology. In Spatial Statistics and Digital Image Analysis. National Research Council, Washington, DC. pp. 181–216.Google Scholar
[2] Cressie, N. A. C. (1991) Spatial Statistics for Spatial Data. Wiley, New York.Google Scholar
[3] Cruz-Orive, L. M. (1980) Best linear unbiased estimators for stereology. Biometrics 36, 595–605.Google Scholar
[4] Cruz-Orive, L. M. (1982) The use of quadrats and test systems in stereology, including magnification corrections. J. Microsc. 125, 89–102.CrossRefGoogle Scholar
[5] Cruz-Orive, L. M. (1989) On the precision of systematic sampling: a review of Matheron's transitive methods. J. Microsc. 153, 315–333.Google Scholar
[6] Cruz-Orive, L. M. (1993) Systematic sampling in stereology. Bull. Int. Statist. Inst. Proc. 49th session, Florence 55, 451–468.Google Scholar
[7] Cruz-Orive, L. M. and Weibel, E. R. (1990) Recent stereological methods for cell biology: a brief survey. Amer. J. Physiol. 258, 148–156.Google Scholar
[8] Davy, P. J. and Miles, R. E. (1977) Sampling theory for opaque spatial specimens. J. R. Statist. Soc. 39, 56–65.Google Scholar
[9] Dehoff, R. T. and Rhines, F. N. (1968) Quantitative Microscopy. McGraw-Hill, New York.Google Scholar
[10] Gual Arnau, X. (1995) Geometría Integral: Curvaturas totales y aplicaciones a la Estereología. Volúmenes de tubos en espacios simétricos. PhD thesis. Dept. Geometría y Topología, Universitat de València (Spain).Google Scholar
[11] Gundersen, H. J. G., Bagger, P., Bendtsen, T. F., Evans, S. M., Korbo, L., Marcussen, N., Møller, A., Nielsen, K., Nyengaard, J. R., Pakkenberg, B., Sørensen, F. B., Vesterby, A. and West, M. J. (1988) The new stereological tools: Disector, fractionator, nucleator and point sampled intercepts and their use in pathological research and diagnosis. APMIS 96, 857–881.Google Scholar
[12] Gundersen, H. J. G., Bendtsen, T. F., Korbo, L., Marcussen, N., Møller, A., Nielsen, K., Nyengaard, J. R., Pakkenberg, B., Sørensen, F. B., Vesterby, A. and West, M. J. (1988) Some new, simple and efficient stereological methods and their use in pathological research and diagnosis. APMIS 96, 379–394.CrossRefGoogle ScholarPubMed
[13] Gundersen, H. J. G. and Jensen, E. B. (1987) The efficiency of systematic sampling in stereology and its prediction. J. Microsc. 147, 229–263.Google Scholar
[14] Jensen, E. B., Baddeley, A. J., Gundersen, H. J. G. and Sundberg, R. (1985) Recent trends in stereology. Int. Statist. Rev. 53, 99–108.Google Scholar
[15] Karlsson, L. M. and Cruz-Orive, L. M. (1991) The new stereological tools in metallography: estimation of pore size in aluminium. J. Microsc. 165, 391–415.Google Scholar
[16] Kendall, D. G. (1974) Foundations of a theory of random sets. In Stochastic Geometry. A Tribute to the Memory of Rollo Davidson. pp 322–376. ed. Harding, E. F. and Kendall, D. G. Wiley, New York.Google Scholar
[17] Mase, S. (1982) Asymptotic properties of stereological estimators of volume fraction for stationary random sets. J. Appl. Prob. 19, 111–126.Google Scholar
[18] Matheron, G. (1975) Random Sets and Integral Geometry. Wiley, New York.Google Scholar
[19] 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. Springer, Berlin. pp. 115–136.Google Scholar
[20] Miles, R. E. and Davy, P. J. (1976) Precise and general conditions for the validity of a comprehensive set of stereological fundamental formulae. J. Microsc. 107, 211–226.Google Scholar
[21] Miles, R. E. and Davy, P. J. (1977) On the choice of quadrats in stereology. J. Microsc. 110, 27–44.Google Scholar
[22] Santaló, L. A. (1976) Integral Geometry and Geometric Probability. Addison-Wesley, Reading, MA.Google Scholar
[23] Stoyan, D., Kendall, W. S. and Mecke, J. (1995) Stochastic Geometry and its Applications. 2nd edn. Wiley, Chichester.Google Scholar
[24] Weibel, E. R. (1979) Stereological Methods. Vol. 1: Practical Methods for Biological Morphometry. Academic Press, London.Google Scholar
[25] Weibel, E. R. (1980) Stereological Methods. Vol. 2: Theoretical Foundations. Academic Press, London.Google Scholar