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. arXiv:1709.07716v2 [stat.ME]

Length-biased data are a particular case of weighted data, which arise in many situations: biomedicine, quality control or epidemiology among others. In this paper we study the theoretical properties of kernel density estimation in the context of length-biased data, proposing two consistent bootstrap methods that we use for bandwidth selection. Apart from the bootstrap bandwidth selectors we suggest a rule-of-thumb. These bandwidth selection proposals are compared with a leastsquares cross-validation method. A simulation study is accomplished to understand the behaviour of the procedures in finite samples.

Wildlife-vehicle collisions on road networks represent a natural problem between human populations and the environment, that affects wildlife management and raise a risk to the life and safety of car drivers. We propose a statistically principled method for kernel smoothing of point pattern data on a linear network when the first-order intensity depends on covariates. In particular, we present a consistent kernel estimator for the first-order intensity function that uses a convenient relationship between the intensity and the density of events location over the network, which also exploits the theoretical relationship between the original point process on the network and its transformed process through the covariate. We derive the asymptotic bias and variance of the estimator, and adapt some data-driven bandwidth selectors to estimate the optimal bandwidth. The performance of the estimator is analysed through a simulation study under inhomogeneous scenarios. We present a real data analysis on wildlife-vehicle collisions in a region of North-East of Spain.

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