Objective Bayesian analysis using modified profile likelihood for the ratio of two log-normal means

Matching priors in principle provide a possible compromise between frequentist and Bayesian and default priors for routine use in Bayesian inference in that posterior probability of the parameter also provides an interpretation as confidence statements. Here we introduce a matching prior for the ratio of two log-normal means which is nontrivial because the derivation of matching prior requires the elicitation of a suitable orthogonal parameterization on the nuisance parameters and the computation of the marginal posterior distribution requires multidimensional integration over the nuisance parameter. Numerical integrations and approximation techniques could be used but they are difficult to use in general. Thus, we derive a matching prior based on a modification of the profile likelihood to avoid the elicitation of priors for the entire parameter and integration on the nuisance parameter. The proposed method is illustrated by real data examples and simulation studies under several configurations.