The distribution of the interpoint distance process of a sequence of pairwise interaction point processes is considered. It is shown that, if the interaction function is piecewise-continuous, then the sequence of interpoint distance processes converges weakly to an inhomogeneous Poisson process under certain sparseness conditions. Convergence of the expectation of the interpoint distance process to the mean of the limiting Poisson process is also established. This suggests a new nonparametric estimator for the interaction function if independent identically distributed samples of the point process are available.

Abstract-The sufficient statistic for performing the likelihood ratio test for pairwise interaction point processes is well-known; however, the evaluation of its performance is a very difficult problem. In this paper it is shown that the distribution of the sufficient statistic can be approximated by the distribution of a Poisson-driven shot-noise random variable, which can be readily computed.Index Terms-Gibbs point process, pairwise interaction point process, Poisson approximation, shot noise.

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The distribution of the interpoint distance process of a sequence of pairwise interaction point processes is considered. It is shown that, if the interaction function is piecewise-continuous, then the sequence of interpoint distance processes converges weakly to an inhomogeneous Poisson process under certain sparseness conditions. Convergence of the expectation of the interpoint distance process to the mean of the limiting Poisson process is also established. This suggests a new nonparametric estimator for the interaction function if independent identically distributed samples of the point process are available.

No abstract