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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: [email protected]
∗∗ 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.
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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 

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