Forward likelihood-based predictive approachfor space–time point processes

Dealing with data from a space–time point process, the estimation of the conditional intensity function is a crucial issue evenif a complete definition of a parametric model is not available. In particular, in case of exploratory contexts or if we want toassess the adequacy of a specific parametric model, some kind of nonparametric estimation procedure could be useful.Often, for these purposes kernel estimators are used and the estimation of the intensity function depends on theestimation of bandwidth parameters. In some fields, like for instance the seismological one, predictive properties ofthe estimated intensity function are pursued. Since a direct ML approach cannot be used, we propose an estimationprocedure based on the subsequent increments of likelihood obtained adding an observation one at a time. Simulatedresults and some applications to statistical seismology are provided. Copyright 2011 John Wiley & Sons, Ltd.