Misspecification issues between competitive spatio-temporal cluster point processes

Spatio-temporal point pattern data often exhibit clustering, which may arise either from self-exciting mechanisms or from latent environmental heterogeneity. Hawkes processes and log-Gaussian Cox processes (LGCPs) represent two widely used but fundamentally different modelling approaches for capturing such patterns. While Hawkes processes assume event-driven triggering, LGCPs model clustering through a latent Gaussian random field acting on the intensity function. In practice, model selection between these alternatives is rarely conducted rigorously, and second-order characteristics are often insufficient to discriminate between them. We present a simulation-based comparative study that systematically evaluates estimation, second-order structure, and predictive performance under model misspecification. In particular, we assess the ability of each model to reproduce the dynamics of data generated under the competing framework. We also analyse real crime data to illustrate how inference and prediction are affected by model choice. Our results underscore the interpretive consequences of model misspecification and highlight key diagnostic limitations when disentangling clustering mechanisms in spatio-temporal processes.