AKaplan-Meier estimators of distance distributions for spatial point processes

Abstract

When a spatial point process is observed through a bounded window, edge effects hamper the estimation of characteristics such as the empty space function $F$, the nearest neighbour distance distribution $G$, and the reduced second order moment function $K$. Here we propose and study product-limit type estimators of $F,G$ and $K$ based on the analogy with censored survival data: the distance from a fixed point to the nearest point of the process is right-censored by its distance to the boundary of the window. The resulting estimators have a ratio-unbiasedness property that is standard in spatial statistics. We show that the empty space function $F$ of any stationary point process is absolutely continuous, and so is the product-limit estimator of $F$. The estimators are strongly consistent when there are independent replications or when the sampling window becomes large. We sketch a CLT for independent replications within a fixed observation window, and asymptotic theory for independent replications of sparse Poisson processes. In simulations the new estimators are generally more efficient than the `border method' estimator but (for estimators of $K$) somewhat less efficient than sophisticated edge corrections.