Bayes inference for the modulated power law process

The Modulated Power Law process has been recently proposed as a suitable model for describing the failure pattern of repairable systems when both renewal-type behaviour and time trend are present. Unfortunately, the maximum likelihood method provides neither accurate confidence intervals on the model parameters for small or moderate sample sizes nor predictive intervals on future observations. This paper proposes a Bayes approach, based on both non-informative and vague prior, as an alternative to the classical method. Point and interval estimation of the parameters, as well as point and interval prediction of future failure times, are given. Monte Carlo simulation studies show that the Bayes estimation and prediction possess good statistical properties in a frequentist context and, thus, are a valid alternative to the maximum likelihood approach. Numerical examples illustrate the estimation and prediction procedures.