Objective Bayesian inference for Birnbaum–Saunders distributions

This paper proposes a Bayesian analysis for the Birnbaum–Saunders distribution and develops noninformative priors for the scale and shape parameters. Probability matching and reference priors are derived for the shape parameter, finding that the second-order matching prior is a high posterior density (HPD) matching prior but not a cumulative density function (CDF) matching prior. For the scale parameter, the second-order matching prior is confirmed to be an HPD matching prior and CDF matching prior but does not match the alternative coverage probabilities up to the second order. The one-at-a-time reference prior and Jeffreys prior satisfy a first-order matching criterion but are not second-order matching priors. The developed priors cause improper posteriors; therefore, modified noninformative priors are suggested that use matching priors with desirable properties. A simulation study demonstrates that these modified matching priors accurately match the target coverage probabilities in a frequentist sense. Two actual examples are provided to illustrate the findings.