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A nonparametric measure of spatial interaction in point patterns. (English) Zbl 0898.62118
Summary: The strength and range of interpoint interactions in a spatial point process can be quantified by the function $$J= (1-G)/(1-F)$$, where $$G$$ is the nearest-neighbour distance distribution function and $$F$$ the empty space function of the process. $$J(r)$$ is identically equal to 1 for a Poisson process; values of $$J(r)$$ smaller or larger than 1 indicate clustering or regularity, respectively. We show that, for a large class of point processes, $$J(r)$$ is constant for distances $$r$$ greater than the range of spatial interaction. Hence both the range and type of interaction can be inferred from $$J$$ without parametric model assumptions. It is also possible to evaluate $$J(r)$$ explicitly for many point process models, so that $$J$$ is also useful for parameter estimation.
Various properties are derived, including the fact that the $$J$$ function of the superposition of independent point processes is a weighted mean of the $$J$$ functions of the individual processes. Estimators of $$J$$ can be constructed from standard estimators of $$F$$ and $$G$$. We compute estimates of $$J$$ for several standard point pattern datasets and implement a Monte Carlo test for complete spatial randomness.

##### MSC:
 62M30 Inference from spatial processes 62G05 Nonparametric estimation
spatial
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