  Modelling the first-order intensity function is one of the main aims in point
process theory, and it has been approached so far from different perspectives.
One appealing model describes the intensity as a function of a spatial
covariate. In the recent literature, estimation theory and several applications
have been developed assuming this model, but without formally checking this
assumption. In this paper we address this problem for a non-homogeneous Poisson
point process, by proposing a new test based on an $L^2$-distance. We also
prove the asymptotic normality of the statistic and we suggest a bootstrap
procedure to accomplish the calibration. Two applications with real data are
presented and a simulation study to better understand the performance of our
proposals is accomplished. Finally some possible extensions of the present work
to non-Poisson processes and to a multi-dimensional covariate context are
detailed.
