Estimation of the pair correlation function of a spatial point process
Rasmus Waagepetersen, Department of Mathematical Sciences, Aalborg University
The so-called pair correlation function is a fundamental spatial point
process characteristic that, given the intensity function, determines
second order moments of the point process. It is of interest in its own
right as a summary of the degree of clustering or regularity of a point
process. Knowledge of it is further needed e.g. to evaluate standard errors
of parameters in a regression model for the intensity function.
Computation of a non-parametric estimate of the pair correlation function
is a typical initial step of a statistical analysis of a spatial point
pattern. Kernel estimates are popular non-parametric estimates and we will
discuss various solutions to the key problem of how to select a suitable
kernel band width.
Especially for clustered point patterns, kernel estimates suffer from bias
for small spatial lags. We introduce a new orthogonal series estimate which
is less biased for clustered point patterns. We consider consistency and
asymptotic normality of the new estimate and also finite sample properties
in a simulation study. Estimates are finally compared in an application to
a data set of tropical rain forest tree locations.