Generalized log likelihood ratio test for non-nested models I understand that if I have two models A and B and A is nested in B then, given some data, I can fit the parameters of A and B using MLE and apply the generalized log likelihood ratio test.  In particular, the distribution of the test should be $\chi^2$ with $n$ degrees of freedom where $n$ is the difference in the number of parameters that $A$ and $B$ have.
However, what happens if $A$ and $B$ have the same number of parameters but the models are not nested? That is they are simply different models.  Is there any way to apply a likelihood ratio test or can one do something else?
 A: The paper Vuong, Q. H. (1989). Likelihood ratio tests for model selection and non-nested hypotheses. Econometrica, 307-333. has the full theoretical treatment and test procedures. It distinguishes between three situations, "Strictly Non-nested Models", "Overlapping Models", "Nested Models", and also examines cases of misspecification. It is therefore no-accident that it finds that for some cases, the test statistic is distributed as a linear combination of chi-squares.  
The paper is not light, neither it proposes an "off-the-shelf" testing procedure. But, for once, its (close to) 3,000 citations speak of its merits, being an inspired combination of classical testing framework and the information-theoretic approach.
A: The Generalised likelihood ratio test DOES NOT work the way you are saying. See for example the following lecture notes:
http://www.maths.manchester.ac.uk/~peterf/MATH38062/MATH38062%20GLRT.pdf
http://www.maths.qmul.ac.uk/~bb/MS_Lectures_12b.pdf
The GLRT is defined for hypothesis of the type:
$$H_0: \theta\in\Theta_0 \,\,\, vs. \,\,\, H_1: \theta\in\Theta_1,$$
where $\Theta_0\cap\Theta_1=\emptyset$ and $\Theta_0\cup\Theta_1=\Theta$.
For the framework you describe, you can compare the models using other tools such as AIC and BIC. Also Bayes factors, if you are willing to go full Bayesian.
