A numerical approach to the Behrens-Fisher problem

The test of the hypothesis of equal means of two normal populations without assumption on the variances is usually referred to as the Behrens-Fisher Problem. Exact similar tests are known not to exist. However, excellent approximately similar “solutions” are readily available. Of these available tests and corresponding critical regions, those due to Welch, Aspin, and Trickett in the 1940s and 1950s come closest to achieving similarity. This article examines numerically the Welch-Aspin asymptotic series and the related Trickett-Welch integral equation formulations of this problem. Through examples, we illustrate that well-behaved tests can deviate from similarity by an almost incredibly small amount. Despite this, with much more extensive computation than was feasible a half-century ago, we can see irregularities which could be an empirical reflection of the known nonexistance of exact solutions.