Semi-parametric estimation of conditional intensity functions in inhomogeneous space-time point processes

Dealing with data coming from a space-time inhomogeneous process, there is often the need of obtaining estimates of the conditional intensity function, without a complete defi nition of a parametric model and so nonparametric estimation is required: isotropic or anisotropic kernel estimates can be used. The properties of the intensities estimated are not always good, expecially in seismological field. We could try to choose the bandwidth in order to have good predictive properties of the estimated intensity function. Since a direct ML approach can not be followed, we use an estimation procedure based on the further increments of likelihood obtained adding a new observation. Similarly to cross validation criterion, we consider additive contributions given by the log-likelihood of the (m+ 1)th observed point given the nonparametric estimation based on the fi rst m observations