Semiparametric estimation of conditional intensity functions for space-time processes

When dealing with data coming from a space time inhomogeneous process, there is often the need of obtaining reliable estimates of the conditional intensity function. According to the field of application, intensity function can be estimated through some assessed parametric model, where parameters are estimated by Maximum Likelihood method. If we are only in an exploratory context or we would like to assess the adequacy of the parametric model, some kind of nonparametric estimation is required. Often, isotropic or anisotropic kernel estimates can be used, e.g. using the Silverman rule for the choice of the windows sizes h (Silverman, 1986). When the purpose of the study is the estimation of h, we could try to choose h in order to have good predictive properties of the estimated intensity function. As it is known, a direct ML approach cannot be followed, since we would obtain degenerate estimates (putting mass only on observed points), unless we use a penalizing function, depending on some smoothing constrai