Kernel Intensity for Space-Time Point Processes with Application to Seismological Problems

Dealing with data coming from a space-time inhomogeneous process, there is often the need of semi-parametric estimates of the conditional intensity function; isotropic or anisotropic multivariate kernel estimates can be used, with windows sizes h. The properties of the intensities estimated with this choice of h are not always good for specific fields of application; we could try to choose h in order to have good predictive properties of the estimated intensity function. Since a direct ML approach cannot be followed, we propose an estimation procedure, computationally intensive, based on the subsequent increments of likelihood obtained adding an observation at time. The first results obtained are very encouraging. Some application in statistical seismology is presented.