Hidden Semi-Markov Modeling for the Estimation of Earthquake Occurrence Rates

The present article aims at the description and application of a discrete-time hidden semi-Markov model in seismology in order to reveal some key features of the earthquake generation process, associated with its causative factor, which is the underlying stress field. The model is based on the assumption that the jump times coincide with the emission times. The parameter space is determined and the empirical estimators of the parameters of the model are calculated in the most general case where the sojourn times are attached to transitions. We focus on the estimation of the semi-Markov kernel that governs the evolution of the hidden semi-Markov chain and the conditional distribution that generates the observations. First, we provide explicit expressions for different important indicators of the underlying semi-Markov chain including hitting times, mean recurrence times and occurrence rates and second we estimate them through a nonparametric method. In addition to the previous results, we investigate the discrete-time intensity of the hitting time (DTIHT) for hidden Markov renewal chains and we provide a statistical estimator.