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Asymptotic properties of estimators for parameters of the Boolean model

Part of: Geometric probability and stochastic geometry Stochastic processes Inference from stochastic processes

Published online by Cambridge University Press:  01 July 2016

Ilya Molchanov  and
Dietrich Stoyan
Ilya Molchanov*
Affiliation:
Freiberg University of Mining and Technology
Dietrich Stoyan*
Affiliation:
Freiberg University of Mining and Technology
*
* On leave from Kiev Technological Institute of the Food Industry.
** Postal address: TU Bergakademie Freiberg, Institut für Stochastik, D-09596 Freiberg, Germany.
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Abstract

This paper considers estimators of parameters of the Boolean model which are obtained by means of the method of intensities. For an estimator of the intensity of the point process of germ points the asymptotic normality is proved and the corresponding variance is given. The theory is based on a study of second-order characteristics of the point process of lower-positive tangent points of the Boolean model. An estimator of the distribution of a typical grain is also discussed.

Keywords

ASYMPTOTIC NORMALITYINTENSITYPOINT PROCESSRANDOM SETPARTICLE SYSTEM

MSC classification

Primary: 62M30: Spatial processes
Secondary: 60D05: Geometric probability and stochastic geometry 60G55: Point processes
Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 26 , Issue 2 , June 1994 , pp. 301 - 323
Copyright
Copyright © Applied Probability Trust 1994 

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Footnotes

This work was supported by the Alexander von Humboldt-Stiftung, Bonn, Germany.

References

Bindrich, U. and Stoyan, D. (1991) Stereology for pores in white bread: statistical analyses for the Boolean model by serial sections. J. Microscopy 162, 231–239.CrossRefGoogle Scholar
Cressie, N. A. C. and Laslett, G. M. (1987) Random set theory and problems of modeling. SIAM Rev. 29, 557–574.CrossRefGoogle Scholar
Cressie, N. A. C. (1991) Statistics for Spatial Data. Wiley, New York etc.Google Scholar
Daley, D. J. and Vere-Jones, D. (1988) An Introduction to the Theory of Point Processes. Springer-Verlag, New York.Google Scholar
Fiksel, T. (1988) Edge-corrected density estimators for point processes. Statistics 19, 67–75.Google Scholar
Girling, A. J. (1982) Approximate variances associated with random configurations of hard spheres. J. Appl. Prob. 19, 588–596.Google Scholar
Hall, P. (1988) Introduction to the Theory of Coverage Processes. Wiley, New York.Google Scholar
Heinrich, L. (1988a) Asymptotic behaviour of an empirical nearest-neighbour distance function for stationary Poisson cluster process. Math. Nachr. 136, 131–148.Google Scholar
Heinrich, L. (1988b) Asymptotic Gaussianity of some estimators for reduced factorial moment measures and product densities of stationary Poisson cluster processes. Statistics 19, 87–106.Google Scholar
Heinrich, L. (1993) Asymptotic properties of minimum contrast estimators for parameters of Boolean models. Metrika 40, 67–94.Google Scholar
Jolivet, E. (1984) Upper bounds of the speed of convergence of moment density estimators for stationary point processes. Metrika 31, 349–360.Google Scholar
Kellerer, A. M. (1985) Counting figures in planar random configurations. J. Appl. Prob. 22, 68–81.Google Scholar
Kellerer, A. M. (1986) The variance of a Poisson process of domains. J. Appl. Prob. 23, 307–321.Google Scholar
Krickeberg, K. (1982) Processus ponctuels en statistique. École d'Été de Probabilitées de Saint-Flour X-1980. Lecture Notes in Mathematics 929, Springer-Verlag, Berlin.CrossRefGoogle Scholar
Lantuejoul, C. and Schmitt, M. (1991). Use of two new formulae to estimate the Poission intensity of a Boolean model. Treizième Colloque GRETSI–Juan-Les-Pins du 16 au 20 Septembre 1991, pp. 1045–1048.Google Scholar
Mase, S. (1982) Asymptotic properties of stereological estimators of volume fraction for stationary random sets. J. Appl. Prob. 19, 111–126.Google Scholar
Matern, B. (1986) Spatial Variation, 2nd edn. Springer-Verlag, Berlin.Google Scholar
Matheron, G. (1975) Random Sets and Integral Geometry. Wiley, New York.Google Scholar
Matthes, K., Kerstan, J. and Mecke, J. (1978) Infinitely Divisible Point Processes. Wiley, Chichester.Google Scholar
Mecke, J. and Stoyan, D. (1980) Formulas for stationary planar fibre processes I-general theory. Math. Operationsforsch. Statist., ser. statist. 12, 267–279.Google Scholar
Molchanov, I. S. (1987) Uniform laws of large numbers for empirical associated functionals of random closed sets. Theory Prob. Appl. 32, 556–559.CrossRefGoogle Scholar
Molchanov, I. S. (1988) Convergence of empirical accompanying functionals of stationary random sets (Russian). Teor. Veroyatnost. i Matem. Statist. 39, 97–99. English translation in: Theory Prob. Math Statist. 39, 107-109.Google Scholar
Molchanov, I. S. (1990) Estimation of the size distribution of spherical grains in the Boolean model. Biom. J. 32, 877–886.Google Scholar
Molchanov, I. S. (1992) Handling with spatial censored observations in statistics of Boolean models of random sets. Biom. J. 34, 617–631.Google Scholar
Molchanov, I. S. (1994) Set-valued estimators for mean bodies related to Boolean models. Statistics. To appear.Google Scholar
Molchanov, I. S. and Stoyan, D. (1994) Directional analysis of fibre processes related to Boolean models. Metrika. To appear.Google Scholar
Preteux, F. and Schmitt, M. (1988) Boolean texture analysis and synthesis. In Image Analysis and Mathematical Morphology, Volume 2: Theoretical Advances, ed. Serra, J., pp. 377–400. Academic Press, New York.Google Scholar
Schmitt, M. (1991). Estimation of the density in a stationary Boolean model. J. Appl. Prob. 28, 702–708.Google Scholar
Schröder, M. (1992) Schätzer für Boolesche Modelle im R2 und R3. Universität Karlsruhe, Diplomarbeit.Google Scholar
Serra, J. (1982) Image Analysis and Mathematical Morphology. Academic Press, London.Google Scholar
Stoyan, D. and Stoyan, H. (1992) Fraktale-Formen-Punktfelder: Methoden der Geometrie-Statistik. Akademie Verlag, Berlin.Google Scholar
Stoyan, D., Kendall, W. S. and Mecke, J. (1987) Stochastic Geometry and Its Applications. Akademie-Verlag, Berlin; Wiley, Chichester.Google Scholar
Stoyan, D., Bertram, U. and Wendrock, H. (1993) Estimation variances for estimators of product densities and pair correlation functions of planar point processes. Ann. Inst. Statist. Math. 45, 211–221.Google Scholar
Weil, W. (1983) Stereology: A survey for geometers. In Convexity and Its Applications, ed. Gruber, P. M. and Wills, J. M., pp. 360–412. Birkhäuser, Basel.Google Scholar
Weil, W. (1988) Expectation formulas and isoperimetric properties for non-isotropic Boolean models. J. Microscopy 151, 235–245.CrossRefGoogle Scholar
Weil, W. (1991) Support densities of random sets. Mathematisches Forschungsinstitut Oberwolfach, Tagungsbericht 53/1991, S.20–21.Google Scholar
Weil, W. and Wieacker, J. A. (1984) Densities for stationary random sets and point processes. Adv. Appl. Prob. 16, 324–346.Google Scholar