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First contact distributions for spatial patterns: regularity and estimation

Published online by Cambridge University Press:  01 July 2016

Martin B. Hansen
Affiliation:
Aalborg University
Adrian J. Baddeley
Affiliation:
University of Western Australia
Richard D. Gill
Affiliation:
University of Utrecht
Corresponding
E-mail address:

Abstract

For applications in spatial statistics, an important property of a random set X in ℝ k 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 F B is absolutely continuous. We give two explicit representations of F B , and additional regularity conditions under which F B is continuously differentiable. A Kaplan-Meier estimator of F B is introduced and its basic properties examined.

Type
Stochastic Geometry and Statistical Applications
Copyright
Copyright © Applied Probability Trust 1999 

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