On the Asymptotic Behaviour of Random Cumulative Hazards

Central limit theorems for functionals of random hazard rates provide a synthetic picture of the global shape of a given hazard rate model. It has been shown that, for the most popular mixture hazard models with completely random mixing measures, the long-term trend and asymptotic variance of the cumulative hazard are not affected by conditioning on observations, so that the behavior of the posterior for large time horizons is systematically determined by the prior choice. In this paper we derive Berry-Ess\'een bounds for prior and posterior CLTs by using a recent technique based on Stein's method for normal approximation of Poisson functionals. The general result is specialized to various specific kernels and it shows that the data enters the second leading term of the prior bound by increasing it.