Frequentist validity of posterior quantiles for a two-parameter exponential family

Berger & Bernardo's (1989) reference priors and Tibshirani (1989) priors are derived for distributions in Bar-Lev & Reiser's (1982) two-parameter exponential family when either the location or the scale parameter is of interest. The reference prior is shown to be a Tibshirani prior. Furthermore, conditions under which a Tibshirani prior is a higher-order matching prior are investigated. When both the parameters are of interest, it is also shown that the reference prior agrees with the prior matching the posterior and frequentist expansions based on the signed square root of log-likelihood ratio. The normal, inverse Gaussian, and gamma distributions are used to illustrate the results. 10.1093/biomet/83.1.55