Goodness-of-fit test for point processes first-order intensity

Modelling the first-order intensity function is one of the main aims in point process theory. An appropriate model describes the first-order intensity as a nonparametric function of spatial covariates. A formal testing procedure is presented to assess the goodness-of-fit of this model, assuming an inhomogeneous Poisson point process. The test is based on a quadratic distance between two kernel intensity estimators. The asymptotic normality of the test statistic is proved and a bootstrap procedure to approximate its distribution is suggested. The proposal is illustrated with two applications to real data sets, and an extensive simulation study to evaluate its finite-sample performance.