Objective Bayesian analysis of common quantiles of normal distributions

Many procedures can assess the equivalence of the means or medians between groups when comparing several independent populations. However, neither the mean nor median captures the entire distribution. We propose a Bayesian approach for the quantiles of several normal distributions, focusing on developing non informative priors to compare the quantiles of multiple populations. Further, reference priors and first- and second-order matching priors are developed. The second-order matching prior does not exist, and the reference priors do not satisfy a first-order matching criterion. In addition, conditions for proper posterior distributions for general priors, including developed priors, are provided. Finally, simulations include a real-world example to demonstrate the proposed approach. The numerical studies indicate that the matching prior performs better than the reference prior and Jeffreys’ prior in terms of matching the frequentist target coverage probabilities, and shows good performance even in small sample sizes. Moreover, the matching prior is not affected by the values of parameters, the number of populations, and does not depend on the values of pth quantile