The Annals of Statistics

Kaplan-Meier estimators of distance distributions for spatial point processes

Adrian Baddeley and Richard D. Gill

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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 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.

Article information

Source
Ann. Statist., Volume 25, Number 1 (1997), 263-292.

Dates
First available in Project Euclid: 10 October 2002

Permanent link to this document
https://projecteuclid.org/euclid.aos/1034276629

Digital Object Identifier
doi:10.1214/aos/1034276629

Mathematical Reviews number (MathSciNet)
MR1429925

Zentralblatt MATH identifier
0870.62028

Subjects
Primary: 62G05: Estimation 62H11: Directional data; spatial statistics 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65]

Keywords
Border correction method dilation distance transform edge corrections edge effects empty space statistic erosion functional delta-method influence function $K$-function local knowledge principle nearest-neighbor distance product integration reduced sample estimator reduced second moment measure sparse Poisson asymptotics spatial statistics survival data

Citation

Baddeley, Adrian; Gill, Richard D. Kaplan-Meier estimators of distance distributions for spatial point processes. Ann. Statist. 25 (1997), no. 1, 263--292. doi:10.1214/aos/1034276629. https://projecteuclid.org/euclid.aos/1034276629


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