Abstract: We introduce a single index model for the intensity of an inhomogeneous spatial point process, relating the intensity function to an unknown function $\mathrm{\rho <!--\; rho\; -->}$ of a linear combination of measurements of a $p$-dimensional spatial covariate process. Such a model extends and generalizes a commonly used model where $\mathrm{\rho <!--\; rho\; -->}$ is known. We derive an estimating procedure for $\mathrm{\rho <!--\; rho\; -->}$ and the coefficient parameters $\mathrm{\beta <!--\; beta\; -->}$ and show consistency and asymptotic normality of estimates of $\mathrm{\beta <!--\; beta\; -->}$ 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.

Key words and phrases: Asymptotic normality, consistency, fast food restaurant data, single-index model, spatial point processes.