Hostname: page-component-78c5997874-j824f Total loading time: 0 Render date: 2024-11-08T21:32:34.879Z Has data issue: false hasContentIssue false
  • English
  • Français

Surprising optimal estimators for the area fraction

Part of: Geometric probability and stochastic geometry Design of experiments Stochastic processes

Published online by Cambridge University Press:  01 July 2016

Katja Schladitz
Katja Schladitz*
Affiliation:
Aalborg University
*
∗ Postal address: Department of Mathematics, Aalborg University, Fredrik Bajers Vej 7E, 9220 Aalborg, Denmark. Email address: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

For a random closed set X and a compact observation window W the mean coverage fraction of W can be estimated by measuring the area of W covered by X. Jensen and Gundersen, and Baddeley and Cruz-Orive described cases where a point counting estimator is more efficient than area measurement. We give two other examples, where at first glance unnatural estimators are not only better than the area measurement but by Grenander's Theorem have minimal variance. Whittle's Theorem is used to show that the point counting estimator in the original Jensen-Gundersen paradox is optimal for large randomly translated discs.

Keywords

Random closed setarea fractioncoverage fractionunbiased estimatorminimum varianceJensen-Gundersen paradoxGrenander's theoremWhittle's theoremsampling designoptimal design

MSC classification

Primary: 60D05: Geometric probability and stochastic geometry 60G55: Point processes
Secondary: 62K05: Optimal designs
Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 31 , Issue 4 , December 1999 , pp. 995 - 1001
Copyright
Copyright © Applied Probability Trust 1999 

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.)

References

Baddeley, A. J. and Cruz-Orive, L. M. (1995). The Rao–Blackwell theorem in stereology and some counterexamples. Adv. Appl. Prob. 27, 2–19.Google Scholar
Cressie, N. A. C. (1991). Statistics for Spatial Data. Wiley, New York.Google Scholar
Grenander, U. (1950). Stochastic processes and statistical inference. Ark. Mat. 1, 195–277.Google Scholar
Jensen, E. B. and Gundersen, H. J. G. (1982). Stereological ratio estimation based on counts from integral test systems. J. Microscopy 125, 51–66.Google Scholar
Näther, W., (1985). Effective Observation of Random Fields. Teubner, Leipzig.Google Scholar
Schladitz, K. (1997). Surprising optimal estimators for the area fraction. Research report 97/24. Department of Mathematics, University of Western Australia.Google Scholar
Stoyan, D., Kendall, W. S. and Mecke, J. (1995). Stochastic Geometry and Its Applications, 2nd edn. John Wiley, Chichester.Google Scholar
Thompson, S. (1992). Sampling. Wiley, Chichester.Google Scholar
Weibel, E. (1979). Stereological Methods, Vol. 1. Practical Methods for Biological Morphometry. Academic Press, London.Google Scholar
Whittle, P. (1973). Some general points in the theory of optimal experimental design. J. Roy. Statist. Soc. B, 35, 123–130.Google Scholar