On A. Wald Test Comparing Two Normal Samples

The well known Wald test comparing the means of two normal samples with unequal variances is of the form: $|\bar x-\bar y/s_2|\geqq\varphi(s_1/s_2)$ (critical zone). A. Wald constructed the function $\varphi$ for which the test is approximately similar with respect to $\sigma_1$ and $\sigma_2$. The present article shows that even under mild conditions on the function $\phi$, Wald's exact similar test is impossible.