Abstract
Many point processes such as earthquakes or volcanic eruptions have incomplete records with the degree of incompleteness varying over time. For these point processes, the number of missing events between each pair of consecutively observed events can be a random variable that may depend on time, effecting the estimation of parameters or hazard. Such incomplete point processes can be modelled by compound renewal processes where the sum of renewal processes is a random variable because of random variable number of missing events. We propose shifted compound Poisson-Gamma and time-dependent shifted compound Poisson-Gamma renewal processes. Since the number of missing events can be regarded as an unobserved process, the proposed renewal processes are introduced to use in the framework of different types of homogeneous and inhomogeneous hidden Markov models to model the time-dependent variable number of missing events between each pair of consecutively observed events of incomplete point processes. Simulation experiments are employed to check the performance of proposed renewal processes with hidden Markov models. We apply the proposed models to the large magnitude explosive volcanic eruptions database to analyze the time-dependent incompleteness and demonstrate how we estimate the completeness of the record and the future hazard rate.