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The comparison of the unknown means of two populations with unknown variances is called the Behrens-Fisher problem, if the populations are assumed to be normal and the ratio of the variances is not known. In this paper a summary of recipes is given to solve this problem in practice, as published in the past 35 years by Banerjee, Fisher and Behrens, Pagurova, Wald and Hajek, Welch, Welch and Aspin, together with two large sample solutions and one solution often used as an approximate one without justification. The solutions are presented mainly in terms of confidence intervals for the difference of the population means. Some remarks are made concerning the lengths of these intervals and the power of the corresponding tests. The solutions in this paper are dependent on the means and the variances of samples drawn from the two populations only. All solutions discussed, except the disqualified approximate one, are robust against violations of the normality assumptions with respect to the populations and they provide, at least asymptotically, good measures for the difference of the population means if the samples are drawn from populations whatsoever with finite second moment.

Numerous studies have documented the robustness of t and F to heterogeneous variances under the restricted condition of equal n’s. Likewise, the distortion of α in the presence of unequal n’s and variances has been demonstrated in both mathematical and empirical studies. Several investigations, however, have shown the Welch technique to be robust to this disturbing situation in the two group case. The present study was addressed to the k group AOV situation. Monte Carlo methods were employed to contrast several procedures with respect to a.) control over Type I errors and b.) power. Results indicate that the generalized Welch technique may be substituted for the AOV when variances are heterogeneous and n’s are unequal.

SUMMARY A brief survey of the recent literature leads to the conclusion that approximately similar tests for the Behrens-Fisher problem retain their practical relevance. The asymptotic expansion of the power function of Welch's test is compared with the asymptotic expansion of the power functions of other asymptotically similar tests. Finally, it is shown that in the class of all translation invariant tests the Welch test is asymptotically uniformly most powerful against one-sided alternatives.

The probabilities of the Type I errors of the Welch approximate-t test and the Aspin-Welch test for the Behrens-Fisher problem have been calculated for selected sets of degrees of freedom and nominal significance levels. The results show satisfactory agreement to the desired level. Thus, in practice, one can just use the usual t-table to carry out the Welch approximate t-test without much loss of accuracy.