Why the F-test in Gaussian linear models is most powerful? For a Gaussian linear model $Y=\mu+\sigma G$ where $\mu$ is assumed to lie in some vector space $W$ and $G$ has the standard normal distribution on $\mathbb{R}^n$, the statistic of the $F$-test for $H_0\colon\{\mu \in U\}$ where $U \subset W$ is a vector space, is an increasing one-to-one function of the deviance statistic:
$$f=\phi\left( 2\log 
\frac{\sup_{\mu \in W, \sigma>0} L(\mu, \sigma | y)}{\sup_{\mu \in U, \sigma>0} L(\mu, \sigma | y)} \right).$$
How can we know that this statistic provides the most powerful test for $H_0$ (maybe after discarding unusual particular cases) ? This doesn't stem from Neyman-Pearson theorem because this theorem asserts that the likelihood-ratio test is the most powerful for point hypotheses $H_0\colon\{\mu=\mu_0, \sigma=\sigma_0\}$ and $H_1\colon\{\mu=\mu_1,\sigma=\sigma_1\}$. 
 A: I have followed this question for some time, hoping that someone with a deeper insight in classical test theory could explain why that $F$-test is not uniformly most powerful in general $-$ just as @cardinal writes in a comment. It is folklore that uniformly most powerful tests can only really be constructed for one-sided hypotheses on univariate parameters, but such a comment doesn't really answer the question.
Example 5.5 in Theoretical Statistics by Cox and Hinkley shows that the $t$-test is a uniformly most powerful similar test for a univariate mean with unknown variance. With a reference to techniques in The Analysis of Variance by Scheffé the same example claims that the $t$-test of a hypothesis on one parameter in the multivariate case is still a uniformly most powerful similar test with the remaining parameters and the variance as nuisance parameters. When the codimension of $U$ is 1, the $F$-test is equivalent to a $t$-test.
Example 5.20, still in Cox and Hinkley, considers one-way ANOVA. It argues that in the case with at least three groups there is no uniformly most powerful similar test of the hypothesis that there is no differences between the groups. This gives the ingredients for showing that the $F$-test is not uniformly most powerful, since for specific alternatives there are more powerful $t$-tests. The $F$-test is, however, the uniformly most powerful invariant test.
So what does similar and invariant then mean? A nested sequence of critical regions for tests of size $\alpha \in [0,1]$ is called similar if the probability of rejecting under the hypothesis is $\alpha$ (for all possible choices of nuisance parameters). The test is invariant if the critical regions are invariant under a group of transformations. For the one-way ANOVA the group is a group of orthogonal transformations. I recommend reading Chapter 5 in Cox and Hinkley for more details. See also Section 2.10 in Scheffé's book on optimum properties of the $F$-test.
