Hostname: page-component-586b7cd67f-2brh9 Total loading time: 0 Render date: 2024-11-25T17:05:59.009Z Has data issue: false hasContentIssue false
  • English
  • Français

Estimating the reduced moments of a random measure

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

Published online by Cambridge University Press:  01 July 2016

Kiên Kiêu  and
Marianne Mora
Kiên Kiêu*
Affiliation:
INRA, France
Marianne Mora*
Affiliation:
Université Paris-X
*
∗ Postal address: Unité de Biométrie, INRA, route de Saint-Cyr, F78026 Versailles Cedex, France. Email address: [email protected]
∗∗ Postal address: U.F.R. de sciences économiques, Université Paris-X, 200 avenue de la république, 92001 Nanterre Cedex, France.
Get access Rights & Permissions [Opens in a new window]

Abstract

We consider a random measure for which distribution is invariant under the action of a standard transformation group. The reduced moments are defined by applying classical theorems on invariant measure decomposition. We present a general method for constructing unbiased estimators of reduced moments. Several asymptotic results are established under an extension of the Brillinger mixing condition. Examples related to stochastic geometry are given.

Keywords

Brillinger mixing conditionedge effectmoment measurenon-parametric estimationpalm distributionpoint processstochastic geometryunbiased estimation

MSC classification

Primary: 60G57: Random measures
Secondary: 60D05: Geometric probability and stochastic geometry 62M99: None of the above, but in this section
Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 31 , Issue 1 , March 1999 , pp. 48 - 62
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

Ambartzumian, R. V. (1990). Factorization Calculus and Geometric Probability (Encyclopedia Math. Appl. 33). CUP, Cambridge, UK.Google Scholar
Baddeley, A. J. (1993). Stereology and survey sampling theory. Bull. Intern. Statist. Inst. 55, 435–449. b3Google Scholar
Barndorff-Nielsen, O. E. (1988). Parametric Statistical Models and Likelihood (Lecture Notes in Statist. 50). Springer, Berlin.Google Scholar
Barndorff-Nielsen, O. E., Blæsild, P. and Eriksen, P.S. (1989). Decomposition and Invariance of Measures, and Statistical Transformation Models (Lecture Notes in Statist. 58). Springer, Berlin.Google Scholar
Bourbaki, N. (1960). Eléments de mathématique: Topologie générale, Chapitres III et IV. Hermann, Paris.Google Scholar
Bourbaki, N. (1963). Eléments de mathématique: Intégration, Chapitres VII et VIII. Hermann, Paris.Google Scholar
Brillinger, D. R. (1975). Statistical inference for stationary point processes. In Stochastic Processes and Related Topics, Vol. 1, ed. Puri, M. L. Academic Press, New York, pp. 55–99.Google Scholar
Hahn, U. and Stoyan, D. (1998). Unbiased stereological estimation of surface area density in gradient surface processes. Adv. Appl. Prob. 30, 904–920.Google Scholar
Jensen, E. B. V. (1998). Local Stereology. World Scientific, Singapore.Google Scholar
Jolivet, E. (1984). Statistique des moments des processus ponctuels stationnaires sur Rd . , Université Paris-sud.Google Scholar
Kallenberg, O. (1983). Random Measures. Akademie, Berlin.Google Scholar
Krickeberg, K. (1974). Moments of point-processes. In Stochastic Geometry (in tribute to Rollo Davidson), ed Harding, E. F. and Kendall, D. G. Wiley, London, pp. 89–113.Google Scholar
Matheron, G. (1975). Random Sets and Integral Geometry. Wiley, New York.Google Scholar
Miles, R. E. (1974). On the elimination of edge effects in planar sampling. In Stochastic Geometry (in tribute to Rollo Davidson), ed Harding, E. F. and Kendall, D. G. Wiley, London, pp. 228–247.Google Scholar
Ohser, J. (1983). On estimators for the reduced second moment measure of point processes. Math. Operationsforsch. Statist. 14, 63–71.Google Scholar
Ohser, J. and Stoyan, D. (1981). On the second-order and orientation analysis of planar stationary point process. Biom. J. 23, 523–533.Google Scholar
Ripley, B. D. (1977). Modelling Spatial Patterns. Wiley, New York.CrossRefGoogle Scholar
Stoyan, D., Kendall, W. S. and Mecke, J. (1995). Stochastic Geometry and its Applications. Wiley, Chichester.Google Scholar