Exact solutions to the Behrens–Fisher Problem: Asymptotically optimal and finite sample efficient choice among

The problem of testing the equality of two normal means when variances are not known is called the Behrens–Fisher Problem. This problem has three known exact solutions, due, respectively, to Chapman, to Prokof’yev and Shishkin, and to Dudewicz and Ahmed. Each procedure has level alpha and power beta when the means differ by a given amount delta, both set by the experimenter. No single-sample statistical procedures can make this guarantee. The most recent of the three procedures, that of Dudewicz and Ahmed, is asymptotically optimal. We review the procedures, and then compare them with respect to both asymptotic efficiency and also (using simulation) in finite samples. Of these exact procedures, based on finite-sample comparisons the Dudewicz–Ahmed procedure is recommended for practical use.