Single-index model for inhomogeneous spatial point processes

We introduce a single index model for the intensity of an inhomogeneous spatial point process, relating the intensity function to an unknown function p of a linear combination of measurements of a p-dimensional spatial covariate process. Such a model extends and generalizes a commonly used model where p is known. We derive an estimating procedure for p and the coefficient parameters β and show consistency and asymptotic normality of estimates of β under some regularity assumptions. We present results of some simulation studies showing the effectiveness of the procedure. Finally, we apply the procedure to a dataset of fast food restaurant locations in New York City.