Second-order analysis of inhomogeneous spatial point processes with proportional intensity functions

We consider pairs of spatial point processes with intensity functions sharing a common multiplicative term. We introduce two novel approaches to estimating the pair correlation function for one of the point processes by treating the other as a baseline so as to account for the unspecified part of the intensity functions. The first approach is based on nonparametric kernel-smoothing, whereas the second approach uses a conditional likelihood estimation approach to fit a parametric model for the pair correlation function. A great advantage of the proposed methods is that they do not require the often difficult estimation of the unspecified part of the intensity functions. We establish the consistency of the resulting estimators and discuss how the parametric estimator can be applied in model diagnostics and inference on regression parameters for the intensity functions. We apply the proposed procedures to two spatial point patterns regarding the spatial distributions of birds in the U.K.'s Peak District in 1990 and 2004.