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I know I can only use the log-likelihoods of two models as selection criterion if they are nested. However, I don't understand this completely. Why isn't it possible to apply this reasoning to non-nested models? I think the higher the log-likelihood the better fits the model the data. I am also aware of other criteria as AIC but this doesn't helps me to understand the issue with the log-likelihoods.

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You can compare the log-likelihoods of two non-nested models. See Cox (1961), "Tests of separate families of hypotheses"), Proc. Fourth Berkeley Symp. on Math. Statist. and Prob., 1 & Cox (1962), "Further results on tests of separate families of hypotheses", JRSS B, 24, 2. But the log-likelihood ratio test statistic doesn't have the asymptotic chi-square distribution with degrees of freedom equal to the difference in the no. estimated parameters that is so convenient when the models compared are nested. See Wilks (1938), "The large-sample distribution of the likelihood ratio for testing composite hypotheses", Ann. Math. Statist., 9, 1.

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    $\begingroup$ If it is comparable, what is needed to be sure to conclude one model is superior? $\endgroup$
    – random_guy
    Feb 13, 2015 at 13:16
  • $\begingroup$ @Scortchi; The comparability you speak can be intended as follow? if the LLF of model A is greater than one of model B, where the number of parameters of A is equal or lower than B, A is better. This is ok? This my question is related: stats.stackexchange.com/questions/546328/… $\endgroup$
    – markowitz
    Oct 4, 2021 at 14:57

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