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Modelling the first-order intensity function is one of the main aims in pointprocess theory, and it has been approached so far from different perspectives.One appealing model describes the intensity as a function of a spatialcovariate. In the recent literature, estimation theory and several applicationshave been developed assuming this model, but without formally checking thisassumption. In this paper we address this problem for a non-homogeneous Poissonpoint process, by proposing a new test based on an $L^2$-distance. We alsoprove the asymptotic normality of the statistic and we suggest a bootstrapprocedure to accomplish the calibration. Two applications with real data arepresented and a simulation study to better understand the performance of ourproposals is accomplished. Finally some possible extensions of the present workto non-Poisson processes and to a multi-dimensional covariate context aredetailed.

Testing first-order intensity model in non-homogeneous Poisson point processes with covariates