Improving the brown-forsythe solution to the generalized behrens-fisher problem

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Abstract
Over two decades ago, Brown and Forsythe (B-F) (1974) proposed an innovative solution to the problem of comparing independent normal means under heteroscedasticity. Since then, their testing procedure has gained in popularity and authors have published various articles in which the B-F test has formed the basis of their research. The purpose of this paper is to point out, and correct, a flaw in the B-F testing procedure. Specifically, it is shown that the approximation proposed by B-F for the null distribution of their test statistic is inadequate. An improved approximation is provided and the small sample null properties of the modified B-F test are studied via simulation. The empirical findings support the theoretical result that the modified B-F test does a better job of preserving the test size compared to the original B-F test.