An improved estimator of the pair correlation function of a spatial point process

SUMMARY The pair correlation function, or two-point correlation, of a spatial point process is a fundamental tool in spatial statistics and astrostatistics, measuring the strength of spatial dependence between points. Interest is focused on the behaviour of this function at short distances, but this is the region in which existing estimators can be particularly unreliable. We propose a new estimator of the pair correlation function based on techniques from stochastic geometry and kernel density estimation. Theory and simulation experiments confirm that the new estimator is far superior to existing estimators, especially at short distances, when the underlying point process is clustered or completely spatially random. Extensions of the estimator are developed for inhomogeneous point processes, for spatially inhibited (negatively correlated) processes and for cases where the form of the pair correlation function is known approximately. We address practical issues including boundary correction, bandwidth selection and data-based choice of technique. Real data examples, of shelling in Ukraine and meningococcal disease in Germany, demonstrate that the new estimator has substantial impact on the interpretation of data.