The asymptotic behavior of tests for normal means based on a variance pre-test

In comparing two normal means, possible inequality of the corresponding variances plays an important role. Attempts to settle this through a preliminatry F-test present new complications: numerical work from literature indicates that the corresponding two-step procedure performs much less satisfactorily than seems to be taken for granted by most textbooks. In the present paper we demonstrate how second order asymptotics can be applied to obtain simple and transparant approximations to size and power of the resulting procedure. This provides both qualitative insight into the behaviour of this procedure, as well as information on the magnitude of the occurring deviations.