Assume I have a set of independent, identically distributed univariate observations $x$ and two hypotheses about how $x$ was generated:
$H_0$: $x$ is drawn from a single Gaussian distribution with unknown mean and variance.
$H_A$: $x$ is drawn from a mixture of two Gaussians with unknown mean, variance and mixing coefficient.
If I understand correctly, these are nested models since the model that $H_0$ represents can be described in terms of $H_A$ if you constrain the parameters of the two Gaussians to be identical or constrain the mixing coefficient to be zero for one of the two Gaussians.
Therefore, it seems like you should be able to use the E-M algorithm to estimate the parameters of $H_A$ and then use Wilks' Theorem to determine whether the likelihood of the data under $H_A$ is significantly greater than that under $H_0$. There's a small leap of faith in the assumption that the E-M algorithm will converge to the maximum likelihood here, but it's one I'm willing to make.
I tried this in a monte carlo simulation, assuming that $H_A$ has 3 more degrees of freedom than $H_0$ (the mean and variance for the second Gaussian and the mixing parameter). When I simulated data from $H_0$, I got a P-value distribution that was substantially non-uniform and enriched for small P-values. (If E-M weren't converging to the true maximum likelihood, the exact opposite would be expected.) What's wrong with my application of Wilks' theorem that's creating this bias?