Noninformative priors for the ratio of scale parameters of the inverse Gaussian distributions

This paper presents an objective Bayesian analysis of the ratio of scale parameters in inverse Gaussian models, which also serves as a basis for testing their homogeneity. We develop noninformative priors for this ratio, focusing on probability matching and reference priors. We show that each prior in the proposed class of second-order matching priors also satisfies the highest posterior density (HPD), cumulative distribution function (CDF), and likelihood ratio (LR) matching criteria, as well as matching alternative coverage probabilities up to second order. Within the class considered, the matching conditions determine the dependence on the first two transformed parameters but leave an arbitrary smooth positive function of the nuisance mean parameters. These conditions therefore characterize a class of priors rather than a unique prior. Furthermore, we show that a specific prior among the three-group reference priors and the one-at-a-time reference prior coincide with a second-order matching prior, whereas the two-group reference prior and Jeffreys’ prior fail to satisfy even the first-order matching criterion. We also establish conditions for the propriety of the posterior distribution under the proposed priors as well as more general priors. The performance of these priors is investigated through simulation studies, and their practical utility is illustrated with two real examples.