Covariance of empirical functionals for inhomogeneous spatial point processes when the intensity has a parametric form

• We estimate the covariance of functionals of inhomogeneous spatial point processes.
• We prove that the kernel-based estimator is consistent and we obtain the optimal asymptotic rate.
• The proposed estimator shows good performance in a simulation study.
• The method is computationally cheaper and less time consuming than existing ones.

This paper is concerned with the problem of estimating covariances of inhomogeneous second-order reweighted stationary spatial point processes when the intensity of the spatial point process has a parametric form. The proposed estimator is based on kernel techniques. It is a very simple and fast estimator which in addition does not require one to model second and higher moments of the spatial point process. Under very mild assumptions, mainly on characteristics of the point process, we prove the mean squared consistency of our estimator. Finally, we show in a simulation study that the kernel-based covariance estimator outperforms existing methods when it is applied to build confidence intervals of the intensity.