Hostname: page-component-586b7cd67f-2brh9 Total loading time: 0 Render date: 2024-11-28T02:00:54.575Z Has data issue: false hasContentIssue false
  • English
  • Français

Asymptotic properties of stereological estimators of volume fraction for stationary random sets

Published online by Cambridge University Press:  14 July 2016

Shigeru Mase
Shigeru Mase*
Affiliation:
Hiroshima University
*
∗Postal address: Faculty of Integrated Arts and Sciences, Hiroshima University, Higashi-Sende-Machi, 1–1–89, Naka-Ku, Hiroshima, 730 Japan.
Get access Rights & Permissions [Opens in a new window]

Abstract

We shall discuss asymptotic properties of stereological estimators of volume (area) fraction for stationary random sets (in the sense of Matheron) under natural and general assumptions. Results obtained are strong consistency, asymptotic normality, and asymptotic unbiasedness and consistency of asymptotic variance estimators. The method is analogous to the non-parametric estimation of spectral density functions of stationary time series using window functions. Proofs are given for areal estimators, but they are also valid for lineal and point estimators with slight modifications. Finally we show that stationary Boolean models satisfy the relevant assumptions reasonably well.

Keywords

STEREOLOGYVOLUME FRACTIONRANDOM SETWINDOW FUNCTIONASYMPTOTIC THEORYSTATIONARITYBOOLEAN MODEL
Type
Research Paper
Information
Journal of Applied Probability , Volume 19 , Issue 1 , March 1982 , pp. 111 - 126
Copyright
Copyright © Applied Probability Trust 1982 

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

Anderson, T. W. (1971) The Statistical Analysis of Time Series. Wiley, New York.Google Scholar
Billingsley, P. (1968) Convergence of Probability Measures. Wiley, New York.Google Scholar
Dehoff, R. T. and Rhines, F. N. (1968) Quantitative Microscopy. McGraw-Hill, New York.Google Scholar
Federer, H. (1969) Geometric Measure Theory. Springer-Verlag, Berlin.Google Scholar
König, D. and Stoyan, D. (1979) On the accuracy of determination of volume and area fraction through lineal analysis. Paper presented at the Fifth International Congress for Stereology, Salzburg, September, 1979.Google Scholar
Mase, S. (1979) A central limit theorem for random fields and its application to random closed set theory. Technical Report No. 8, Statistical Research Group, Hiroshima University.Google Scholar
Matheron, G. (1975) Random Sets and Integral Geometry. Wiley, New York.Google Scholar
Miles, R. E. (1978) The importance of proper model specification in stereology. In Proceedings of the Buffon Bicentenary Symposium, Lecture Notes in Biomathematics 23, Springer-Verlag, Berlin, 115–136.Google Scholar
Nguyen, X. X. and Zessin, H. (1979) Ergodic theorems for spatial processes. Z. Wahrscheinlichkeitsth. 48, 133–158.CrossRefGoogle Scholar
Stoyan, D. (1979) On the accuracy of lineal analysis. Biometrical J. 21, 439–449.CrossRefGoogle Scholar
Suwa, N. (1977) Quantitative Stereology (in Japanese). Iwanami Book Company, Tokyo.Google Scholar
Underwood, E. E. (1970) Quantitative Stereology. Addison-Wesley, Reading, Ma.Google Scholar