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 the 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 dierent types of point processes. In particular it detects a clear dierence between the Poisson process and the Baddeley-Silverman cell process whereas the K-functions for these processes coincide. Key words: stationary point process, edge correction, third factorial moment measure, third reduced moment measure..