# 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?

• Are $\sigma_1,\sigma_2$ known? – StubbornAtom Oct 23 '18 at 5:45

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 ).