Estimating the hazard of rupture using uncertain occurrence times of paleoearthquakes

When a renewal process is fitted to paleoearthquake occurrence data, it is more informative to use the data including the two interval ends before the first event to the beginning of observation period and after the last event till the present time because the sample size is usually small. Our main concern is analysis of the data in which some occurrence times of events themselves are uncertain and given by intervals or some distributions. Specifically, we consider a Bayesian inference where each uncertainty is interpreted as a prior distribution associated with likelihood of a renewal process model. Integration of the posterior with respect to the occurrence times and its maximization with respect to the parameters of the renewal process model are implemented in order to estimate the parameters and further to compare the goodness of fit of other competing renewal process models. Thus the posterior of the selected model provides a reasonable estimate of distribution of uncertain occurrence times associated with the fitted renewal process model. Finally, in a case where occurrence time of the last event is uncertain, a practical method for the assessment of current and future hazard of the forthcoming rupture is provided.