A Third Order Point Process Characteristic

Second order characteristics, in particular Ripley's K-function, are widely used for the statistical analysis of point patterns. We examine a third order analogue, namely the mean number T(R) of R-close triples of points per unit area. Equivalently this is the expected number of R-close point pairs in an R-neighbourhood of a typical point. Various estimators for this function are proposed and compared, and we give an explicit formula for the isotropic edge correction. The theoretical value of T seems to be unobtainable for most point process models apart from the homogeneous Poisson process. However, simulation studies show that the function T discriminates well between diierent types of point processes. In particular it detects a clear diierence between the Poisson process and the Baddeley-Silverman cell process whereas the K-functions for these processes coincide.