On Bayesian Nonparametric Estimation of Smooth Hazard Rates with a View to Seismic Hazard Assessment

WORKING PAPER R 38-05, DIPARTIMENTO DI SCIENZE SOCIALI, COGNITIVE E QUANTITATIVE, UNIVERSITA' DI MODENA E REGGIO EMILIA, REGGIO EMILIA, ITALY.Nonparametric inference on the hazard rate is an alternative to density estimation for positive variables which naturally deals with right censored observations. It is a classic topic of survival analysis which is here shown to be of interest in the applied context of seismic hazard assessment. This paper puts forth a new Bayesian approach to hazard rate estimation, based on building the prior hazard rate as a convolution mixture of a probability density with a compound Poisson process. The resulting new class of nonparametric priors is studied in view of its use for Bayesian inference: first, conditions are given for the prior to be well defined and to select smooth distributions; then, a procedure is developed to choose the hyperparameters so as to assign a constant expected prior hazard rate, while controlling prior variability; finally, an MCMC approximation of the posterior distribution is found. The proposed algorithm is implemented for the analysis of some Italian seismic event data and a possible adjustment to a well established class of prior hazard rates is discussed in some detail.