Objective Bayesian analysis for the ratio of shape parameters in generalized half-normal distributions

The generalized half-normal distribution is notable for its flexibility in modeling diverse hazard rate shapes, including ones that are monotonically increasing or decreasing, and bathtub forms, determined by the value of the shape parameter. To facilitate Bayesian comparative analyses of these shape parameters, we have proposed noninformative priors for the ratio of shape parameters within generalized half-normal distributions. We derived probability matching priors and reference priors, identifying a second-order matching prior that satisfies all specified matching criteria. Our findings show that both the two-group and three-group reference priors meet the first-order matching criterion, whereas Jeffreys' prior does not. However, the one-at-a-time reference prior successfully satisfies the stricter second-order matching criterion. Additionally, we established conditions ensuring posterior propriety under general priors, particularly highlighting the derived noninformative priors. A simulation study demonstrated that the one-at-a-time reference prior achieves accurate alignment with target frequentist coverage probabilities. Finally, we provided two real-world examples to illustrate and reinforce our theoretical results.