Would anybody be able to explain to me why the likelihood function $L$ can not be used to compare nonnested models whereas the AIC, which is a simple function of $L$ ($-2\log L+2k$ where $k$ is the number of parameters), can?


In particular I am wondering how two models such as gamma and lognormal, with completely different likelihood functions can then be compared because of a correction of the number of parameters. I believe that these likelihood functions are on different scales because of different normalising constants and are therefore quite different.


It is my understanding (I may be quite wrong) that the AIC represents how close a model is to minimising the "distance" (entropy) to the true model. The likelihood function represents a different distance but a distance all the same. Why the sudden jump from nested to nonnested?


Thanks in advance!



use of akaike information criterion with nonnested models
In theory, either can be used, even when the models are different. The idea is that you use the Kullback-Liebler divergence to choose between the models. You can estimate this by taking the sample log likelihood, and divide by the sample size. It is known that this can be biased in small samples with a bias proportional to the number of parameters, so the AIC is an attempt to adjust for this bias. In large samples, the correction (when divided by the sample size) becomes negligible. 


Note that if you want to any hypothesis testing, the asymptotic statistics for non-nested models is different. The asymptotics were worked out by [Vuong](https://en.wikipedia.org/wiki/Vuong%27s_closeness_test).


