Estimating the J function without edge correction

The interaction between points in a spatial point process can be measured by its empty space function F, its nearest-neighbour distance distribution function G, and by combinations such as the J function J = (1 - G)/(1 - F). The estimation of these functions is hampered by edge effects: the uncorrected, empirical distributions of distances observed in a bounded sampling window W give severely biased estimates of F and G. However, in this paper we show that the corresponding uncorrected estimator of the function J = (1- G)/(1 - F) is approximately unbiased for the Poisson case, and is useful as a summary statistic. Specifically, consider the estimate (J) over cap (W) of J computed from uncorrected estimates of F and G. The function J(W)(r), estimated by (J) over cap (W), possesses similar properties to the J function, for example J(W)(r) is identically 1 for Poisson processes. This enables direct interpretation of uncorrected estimates of J, something not possible with uncorrected estimates of either F, G or K. We propose a Monte Carte test for complete spatial randomness based on testing whether J(W)(r) = 1. Computer simulations suggest this test is at least as powerful as tests based on edge corrected estimators of J.