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Comparisons and asymptotics for empty space hazard functions of germ-grain models

Part of: Geometric probability and stochastic geometry Stochastic processes

Published online by Cambridge University Press:  01 July 2016

Günter Last  and
Ryszard Szekli
Günter Last*
Affiliation:
Karlsruher Institut für Technologie
Ryszard Szekli*
Affiliation:
University of Wrocław
*
∗ Postal address: Institut für Stochastik, Karlsruher Institut für Technologie, 76128 Karlsruhe, Germany. Email address: guenter.last@kit.edu
∗∗ Postal address: Mathematical Institute, pl. Grunwaldzki 2/4, 50-384 Wrocław, Poland.
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Abstract

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We study stochastic properties of the empty space for stationary germ-grain models in Rd; in particular, we deal with the inner radius of the empty space with respect to a general structuring element which is allowed to be lower dimensional. We consider Poisson cluster and mixed Poisson germ-grain models, and show in several situations that more variability results in stochastically greater empty space in terms of the empty space hazard function. Furthermore, we study the asymptotic behaviour of the empty space hazard functions at 0 and at ∞.

Keywords

Germ-grain modelempty space hazardrelative distancepoint processPoisson cluster processmixed Poisson processhazard rate ordering

MSC classification

Primary: 60D05: Geometric probability and stochastic geometry
Secondary: 60G55: Point processes

Information

Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 43 , Issue 4 , December 2011 , pp. 943 - 962
Copyright
Copyright © Applied Probability Trust 2011 

Footnotes

This work was supported by MNiSW Research Grant N N201 394137.

References

Baddeley, A. and Gill, R. D. (1994). The empty space hazard of spatial pattern. Preprint 845, Department of Mathematics, University o Utrecht.Google Scholar
Baddeley, A. and Gill, R. D. (1997). Kaplan-Meier estimators of distance distributions for spatial point processes. Ann. Statist. 25, 263–292.Google Scholar
Bordenave, C. and Torrisi, G. L. (2007). Large deviations of Poisson cluster processes. Stoch. Models 23, 593–625.Google Scholar
Chiu, S. N. and Stoyan, D. (1998). Estimators of distance distributions for spatial patterns. Statist. Neerlandica 52, 239–246.Google Scholar
Daley, D. J. and Vere-Jones, D. (2003). An Introduction to the Theory of Point Processes, Vol. I, 2nd edn. Springer, New York.Google Scholar
Esary, J. D., Proschan, F. and Walkaup, D. W. (1967). Association of random variables, with applications. Ann. Math. Statist. 38, 1466–1474.Google Scholar
Hall, P. (1988). Introduction to the Theory of Coverage Processes. John Wiley, New York.Google Scholar
Hansen, M. B., Baddeley, A. J. and Gill, R. D. (1999). First contact distributions for spatial patterns: regularity and estimation. Adv. Appl. Prob. 31, 15–33.Google Scholar
Hug, D. and Last, G. (2000). On support measures in Minkowski spaces and contact distributions in stochastic geometry. Ann. Prob. 28, 796–850.Google Scholar
Hug, D., Last, G. and Weil, W. (2002). Generalized contact distributions of inhomogeneous Boolean models. Adv. Appl. Prob. 34, 21–47.Google Scholar
Hug, D., Last, G. and Weil, W. (2002). A survey on contact distributions. In Morphology of Condensed Matter (Lecture Notes Phys. 600), eds Mecke, K. and Stoyan, D., Springer, Berlin, pp. 317–357.Google Scholar
Last, G. and Holtmann, M. (1999). On the empty space function of some germ–grain models. Pattern Recognition 32, 1587–1600.Google Scholar
Last, G. and Schassberger, R. (1998). On the distribution of the spherical contact vector of stationary grain models. Adv. Appl. Prob. 30, 36–52.Google Scholar
Molchanov, I. (1997). Statistics of the Boolean model for Practitioners and Mathematicians. John Wiley, Chichester.Google Scholar
Molchanov, I. (2005). Theory of Random Sets. Springer, London.Google Scholar
Müller, A. and Stoyan, D. (2002). Comparison Methods for Stochastic Models and Risks. John Wiley, Chichester.Google Scholar
Schneider, R. (1993). Convex Bodies: The Brunn-Minkowski Theory (Encyclopedia Math. Appl. 44). Cambridge University Press.Google Scholar
Schneider, R. and Weil, W. (2008). Stochastic and Integral Geometry. Springer, Berlin.Google Scholar
Stoyan, D. (1983). Comparison Methods for Queues and Other Stochastic Models. John Wiley, Chichester.Google Scholar
Stoyan, D., Kendall, W. S. and Mecke, J. (1995). Stochastic Geometry and Its Applications, 2nd edn. John Wiley, Chichester.Google Scholar
Stoyan, H. and Stoyan, D. (1980). On some partial orderings of random closed sets. Math. Operationsforsch. Statist. Ser. Optimization 11, 145–154.Google Scholar
Szekli, R. (1995). Stochastic Ordering and Dependence in Applied Probability (Lecture Notes Statist. 97). Springer, New York.Google Scholar
Van Lieshout, M. N. M. and Baddeley, A. J. (1996). A nonparametric measure of spatial interaction in point patterns. Statist. Neerlandica 50, 344–361.Google Scholar