is also unknown, then the problem of testing the hypotheses (8.6.1) or the hypotheses (8.6.8) becomes very difficult. Even the likelihood ratio statistic A has no known distribution. This problem is known as the Behrens-Fisher problem. Some simulation methods for the Behrens-Fisher problem will be described in Chapter 11 (Examples 11.1.4 and 11.5.9) various other test procedures have been proposed, but most of them have been the subject of controversy in regard to their appropriateness or usefulness. The most popular of the proposed methods was developed in a series of articles by Welch (1938,1947,1951).Welch proposed using the statistic