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Kaplan-Meier estimators of distance distributions for spatial point processes. (English) Zbl 0870.62028
Summary: 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.

MSC:
62G05 Nonparametric estimation
62M30 Inference from spatial processes
62H11 Directional data; spatial statistics
62G20 Asymptotic properties of nonparametric inference
60D05 Geometric probability and stochastic geometry
Software:
spatial
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