On Behrens-Fisher Solutions

Abstract

SUMMARY This paper builds on recent work of Barnard, Sprott and Farewell, and Duong and Shorrock to give an overview of some practical solutions to the two-means Behrens-Fisher problem. The classical fiducial solution and the WelchSatterthwaite approximate frequentist solution are reviewed, then Bayes solutions of Barnard and finally an empirical Bayes solution of Duong and Shorrock, which we recommend if a simple summary (interval estimate orp-value) of the inference is desired. An empirical Bayes interpretation of the work of Barnard for p-values, and of Sprott and Farewell for confidence intervals, is described. All numerical work is easy to implement on certain programmable calculators (Hewlett-Packard HP41 with Advantage Pac, BP42, BP28, HP48, etc.), without the use of tables. All solutions discussed are applied to data given by Sprott and Farewell, Brownlee and to an example used by Barnard to illustrate some of the concepts involved.