Model selection, LRT test

Dear list,

If I do not make a mistake, use of Likelihood ratio test is precluded
when two models have the same number of degree of freedom.
Is there a way to test which one is the best when both are close in
AIC value (difference<  5) or do I have to conclude that they are
"equivalent" ?

In my understanding the LRT makes sense *only* if the two models
are *nested*; one (the ``null'' or ``restricted'' model is a sub-model
of the ``full'' or ``alternative'' model.  Whence there *has* to be a
difference in the number of parameters.

I do not know if it is possible to do formal inference as to whether
one model ``improves'' over the other in settings in which the
models are not nested.

     cheers,

         Rolf Turner