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 neighbor 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.
|Annals of Statistics
|Published - 1997