Bayesian analysis of repairable systems with modulated power law process

As a compromise between nonhomogeneous Poisson process and renewal process, the modulated power law process is more appropriate to model the failures of repairable systems. In this article, objective Bayesian methods are proposed to analyze the modulated power law process. Seven reference priors, one of which is also the Jeffreys prior, are derived. However, only four of them are taken into consideration because of their practicality. Propriety of the posterior densities considering the four reference priors is proved. Predictive distribution of the future failure time is obtained additionally. For the purpose of comparison, the simulation work and real data analysis are carried out based on both the objective Bayesian and maximum likelihood approaches, which show that the objective Bayesian estimation and prediction have much better statistical properties in a frequentist context, and outperforms the maximum likelihood method even with small or moderate sample sizes.