On the Behrens-Fisher Problem: A Review

Abstract

The Behrens-Fisher problem arises when one seeks to make inferences about the means of two normal populations without assuming the variances are equal. This paper presents a review of fundamental concepts and applications used to address the Behrens-Fisher problem under fiducial, Bayesian, and frequentist approaches. Methods of approximations to the Behrens-Fisher distribution and a simple Bayesian framework for hypothesis testing are also discussed. Finally, a discussion is provided for the use of generalized p values in significance testing of hypotheses in the presence of nuisance parameters. It is shown that the generalized p values based on a frequentist probability for the Behrens-Fisher problem are numerically the same as those from the fiducial and Bayesian solutions. A table for tests of significance is also included.