The Behrens-Fisher problem revisited: A Bayes-frequentist synthesis

The Behrens-Fisher problem concerns the inference for the difference between the means of two normal populations whose ratio of variances is unknown. In this situation, Fisher’s fiducial interval differs markedly from the Neyman-Pearson confidence interval. A prior proposed by Jeffreys leads to a credible interval that is equivalent to Fisher’s solution, but carries a different interpretation. The authors propose an alternative prior leading to a credible interval whose asymptotic coverage probability matches the frequentist coverage probability more accurately than Jeffreys’ interval. Their simulation results indicate excellent matching even in small samples.