Noninformative priors for the nested design

This paper considers noninformative priors for three-stage nested designs. It turns out that the noninformative prior given by Li and Stern (1997) is the one-at-a-time reference prior satisfying a second-order matching criterion when either the variance ratio or linear combinations of the means is of interest. Moreover, it is a joint probability matching prior when both the variance ratio and linear combinations of the means are of interest. These priors are compared with Jeffreys' prior in light of how accurately the coverage probabilities of Bayesian credible intervals match the corresponding frequentist coverage probabilities.