First contact distributions for spatial patterns: regularity and estimation

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.