Objective Bayesian tests for Fieller–Creasy problem

From a statistical perspective, the Fieller–Creasy problem which involves inference about the ratio of two normal means has been quite challenging. In this study, we consider some solutions to this problem, based on an objective Bayesian model selection procedure. First, we develop the objective priors for testing the ratio of two normal means, based on measures of divergence between competing models. We then propose the intrinsic priors and the fractional priors for which the Bayes factors and model selection probabilities are well defined. In addition, we prove that the Bayes factors based on divergence-based priors, as well as intrinsic and fractional priors, are consistent for large sample sizes. Finally, we derive the Bayesian reference criterion from the Bayesian decision theory framework, based on the intrinsic discrepancy loss function. The behaviors of the Bayes factors are compared by undertaking a simulation study and using a case study example.