Hostname: page-component-cd9895bd7-mkpzs Total loading time: 0 Render date: 2024-12-28T00:55:18.106Z Has data issue: false hasContentIssue false
  • English
  • Français

First contact distributions for spatial patterns: regularity and estimation

Part of: Multivariate analysis Nonparametric inference Geometric probability and stochastic geometry

Published online by Cambridge University Press:  01 July 2016

Martin B. Hansen ,
Adrian J. Baddeley  and
Richard D. Gill
Martin B. Hansen*
Affiliation:
Aalborg University
Adrian J. Baddeley*
Affiliation:
University of Western Australia
Richard D. Gill*
Affiliation:
University of Utrecht
*
∗ Postal address: Department of Mathematical Sciences, Aalborg University, Fredrik Bajers Vej 7 E, DK 9220 Aalborg Ø, Denmark. Email address: mbh@math.auc.dk
∗∗ Postal address: Department of Mathematics, University of Western Australia, Nedlands WA 6907, Australia.
∗∗∗ Postal address: Mathematical Institute, University of Utrecht, Budapestlaan 6, 3584 CD Utrecht, The Netherlands.
Get access Rights & Permissions [Opens in a new window]

Abstract

For applications in spatial statistics, an important property of a random set X in ℝk is its first contact distribution. This is the distribution of the distance from a fixed point 0 to the nearest point of X, where distance is measured using scalar dilations of a fixed test set B. We show that, if B is convex and contains a neighbourhood of 0, the first contact distribution function FB is absolutely continuous. We give two explicit representations of FB, and additional regularity conditions under which FB is continuously differentiable. A Kaplan-Meier estimator of FB is introduced and its basic properties examined.

Keywords

Empty space functionconvex test setsrandom closed setsstochastic geometryspatial statisticscoarea formulaproduct limit estimateshazard rate

MSC classification

Primary: 60D05: Geometric probability and stochastic geometry 62H11: Directional data; spatial statistics
Secondary: 62G05: Estimation
Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 31 , Issue 1 , March 1999 , pp. 15 - 33
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

Andersen, P. K., Borgan, Ø., Gill, R. D. and Keiding, N. (1993). Statistical Models Based on Counting Processes. Springer, New York.Google Scholar
Baddeley, A. (1977). Integrals on a moving manifold and geometrical probability. Adv. Appl. Prob. 9, 588–603.Google Scholar
Baddeley, A. J. and Gill, R. D. (1993). Kaplan–Meier estimators of interpoint distance distributions for spatial point processes. Research Report, BS-R9315, Centrum voor Wiskunde en Informatica, Amsterdam.Google Scholar
Baddeley, A. J. and Gill, R. D. (1997). Kaplan–Meier estimators of interpoint distance distributions for spatial point processes. Ann. Statist. 25, 263–292.Google Scholar
Cressie, N. A. C. (1991). Statistics for Spatial Data. Wiley, New York.Google Scholar
Diggle, P. J. (1983). Statistical Analysis of Spatial Point Patterns. Academic Press, London.Google Scholar
Evans, L. C. and Gariepy, R. F. (1992). Measure Theory and Fine Properties of Functions. CRC Press, Boca Raton, FA.Google Scholar
Federer, H. (1969). Geometric Measure Theory. Springer, New York.Google Scholar
Gill, R. D. (1994). Lectures on survival analysis. In Lectures on Probability Theory, Ecole d'Eté de Probabilités de Saint-Flour XXII 1992, ed. Bernard, P. (Lecture Notes in Math. 1581). Springer, New York.Google Scholar
Hansen, M. B., Baddeley, A. J. and Gill, R. D. (1996). Kaplan–Meier type estimators for linear contact distributions. Scand. J. Statist. 23, 129–155.Google Scholar
Hardt, R. and Simon, L. (1986). Seminar on Geometric Measure Theory. Birkhäuser, Basel.Google Scholar
Kendall, M. G. and Moran, P. A. P. (1963). Geometrical Probability (Griffin's Statistical Monographs and Courses 10). Griffin, London.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: a tribute to the memory of Rollo Davidson, ed. Harding, E. F. and Kendall, D. G. Wiley, New York, pp. 228–247.Google Scholar
Morgan, F. (1988). Geometric Measure Theory: A Beginners Guide. Academic Press, Boston.CrossRefGoogle Scholar
Ripley, B. D. (1988). Statistical Inference for Spatial Processes. Cambridge University Press.Google Scholar
Serra, J. (1982). Image Analysis and Mathematical Morphology, Vol. 1. Academic Press, London.Google Scholar
Simon, L. M. (1984). Lectures on geometric measure theory. In Proc. Centre for Mathematical Analysis, Australian National University, Vol. 3. Australian National University, Canberra.Google Scholar
Stoyan, D., Kendall, W. S. and Mecke, J. (1995). Stochastic Geometry and Its Applications, 2nd edn. Wiley, Chichester.Google Scholar