Advances in Applied Probability

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

Günter Last and Ryszard Szekli

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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 ∞.

Article information

Adv. in Appl. Probab., Volume 43, Number 4 (2011), 943-962.

First available in Project Euclid: 16 December 2011

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65]
Secondary: 60G55: Point processes

Germ-grain model empty space hazard relative distance point process Poisson cluster process mixed Poisson process hazard rate ordering


Last, Günter; Szekli, Ryszard. Comparisons and asymptotics for empty space hazard functions of germ-grain models. Adv. in Appl. Probab. 43 (2011), no. 4, 943--962. doi:10.1239/aap/1324045693.

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