Why is the ML ratio test of two normal means with different variances impossible? 
Say I have IID samples from possibly different means and exclusive variances, $X_{1i} \sim N(\mu_1,\sigma_1^2), X_{2j} \sim N(\mu_2,\sigma_2^2)$, where $(i = 1, \cdots, m, j = 1, \cdots, n)$, and $\sigma_1 \neq \sigma_2$.
Test : $H_0 : \mu_1 = \mu_2$ vs. $H_1 : \mu_1 \neq \mu_2$.

According to my professor, there is no maximum likelihood ratio test with significance level $\alpha$ for this model.
We exactly know that there can be test (t-test) for different mean-same variance test, and F-test for the comparison of variances from possibly different mean, and different variance model.
But I cannot find the formal proof for this. He mentioned that this fact can be driven from techniques of Harmonic Analysis, but is there anyone to exactly show what it says?
 A: Hi: For a test to be optimal,  a necessary condition ( there are others ) is that the test statistic used for the test has to be a pivotal quantity. Being a pivotal quantity means that the statistic has to have a distribution that's independent of all the parameters of the distribution except for the one 
being tested.
So, in the simple case where say, there's one mean being tested and normality is assumed, it's possible to divide the parameter being tested, $\beta$, by the variance ( or the estimated variance ) and come up with a test-statistic that is a  pivotal quantity. ( because it's normal zero one under the null ).
In the case of 2 means and 2 different variances, it's not possible to divide the numerator by something and obtain a pivotal quantity. I've never seen a proof of the impossibility of this in a statistics text so your professor is probably correct that the proof uses harmonic analysis. But I hope  that I atleast somewhat explained what is trying to be proven so that you know what to look for when searching for a proof. Googling for Sattherwaite's Approximation may help during your search also. Good luck.
P.S: Another necessary condition ( and maybe sufficient, I think ) for optimality of the test is that the test statistic should be able to be derived using the neyman pearson lemma. But, as far what your professor is talking about in regard to harmonic analysis, my guess is that he's referring to the pivotal quantity part of the necessary condition.
