Nonparametric Bayesian inference in renewal recurrence models

Despite the large number of studies on the time-dependent hazard a commonly accepted model does not exist yet, but there are some more shared conjectures as the dependence on the time elapsed since the last occurrence, that is, the idea that after each event the probability of the future process may be the same. From the probabilistic point of view this means to consider renewal processes. They are appropriate to model large earthquakes after which one can assume that stress accumulation process restarts. Statistically this implies that the times between large seismic events can be considered as realizations of independent, identically distributed random variables, the interoccurrence times. Various probability distributions have been considered in the literature for these times. When satisfactory results in terms of fitting were obtained, then one looked for physical interpretations of the underlying process; for instance, distributions with decreasing hazard can support the idea that there are strong interactions among neighbouring fault segments. We prefer the opposite way of dealing with the problem: we do not make any assumption on the functional form of the distribution of the interoccurrence time and strengthen the probabilistic methodology by considering such a distribution as a random function modelled by a Polya process. The approach followed for estimation is Bayesian, semi-parametric; we just assume that the prior expectation of the unknown density function is chosen in the class of the generalized gamma distributions for its extremely great flexibility. Renewal models have to be applied in intraplate regions for global hazard assessment and for decade-long forecasts.