Recommendations on cubature parameter selection for likelihood approximation in three-dimensional Poisson point processes

Likelihood-based inference for three-dimensional Poisson point processes requires numerical approximation of the integral term in the log-likelihood through a cubature scheme algorithm. The quality of this approximation, and hence the accuracy of the resulting statistical inference, depends on a small set of tuning parameters controlling the cubature construction. Despite their practical importance, the literature provides little guidance on how these parameters should be selected in order to obtain reliable first-order inference. This paper addresses this issue for purely spatial three-dimensional Poisson point process models. We formalize the cubature scheme in R3 and conduct an extensive simulation study across multiple Poisson process scenarios, process sizes, and cubature configurations. Cubature settings are evaluated by combining parameter mean squared error with a second-order diagnostic based on the three-dimensional inhomogeneous K-function and the Global Envelope Test. The simulation results are then aggregated into empirically grounded practical recommendations for selecting the dummy-point ratio, the tessellation resolution, and the dummy-point layout. Finally, a real three-dimensional spatial application illustrates how cubature choices consistent with these recommendations can lead to stable parameter estimates, reliable fitted intensities, and satisfactory diagnostic performance.