Kaplan-Meier type distributions for linear contact distributions

Abstract

The linear contact distribution function is shown to be continuously dierentiable for any stationary random closed set which implies the existence of a continuous density and hazard rate Moreover it is proved that the density is monotone decreasing When the linear contact distribution function is estimated from observations in a bounded window the distance to the set of interest from a xed point in a given linear direction is rightcensored by its distance to the boundary of the window We develop a KaplanMeier type estimator for the linear contact distribution function and hazard rate We show that the new estimator has a ratiounbiasedness property and that it is an absolutely continuous distribution function A CLT is derived for independent replications within a xed observation window The techniques are applied to the analysis of spatial patterns in acid milk The feature of replication of the images and the CLT for the estimator give condence bounds on the estimator This is used to discriminate between dierent kinds of heat treatments