Intrinsic Bayes factor approach to a test for the power law process

Abstract The versatile new criterion called the intrinsic Bayes factor (IBF), introduced by Berger and Pericchi [J. Amer. Statist. Assoc. 91 (1996) 109–122], has made it possible to perform model selection and hypotheses testing using standard (improper) noninformative priors in a variety of situations. In this paper, we use their methodology to test several hypotheses regarding the shape parameter of the power law process, which has been widely used to model failure times of repairable systems. Assuming that we have data from the process according to the time-truncation sampling scheme, we derive the arithmetic IBFs using four default priors, including the reference and Jeffreys priors. We establish the frequentist probability matching properties of these priors. We also identify two priors that are justifiable under both time-truncation and failure-truncation schemes, so that the IBFs for both schemes can be unified. Deducing the intrinsic priors of a certain canonical form, as the time of truncation tends to infinity, we show that the arithmetic IBFs correspond asymptotically to actual Bayes factors. We also discuss the expected IBFs, which are useful with small samples. We then use these results to analyze an actual data set on the interruption times of a transmission line, summarizing our results under the default priors.