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.