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For example AIC penalises the number of parameters some goodness of fit tests assess the fit on the tails or the shoulders of the distribution and some other evaluate the predictive performance of the models in question. You should correct for small sample sizes if you use the AIC with small sample sizes by using the AICc statistic. Bayesian Correct Number Of Components In Gmm According To Bic And Aic Plots Cross Validated We can see that both in the Akaike and Likelihood ratio test essentially I compare the likelihood functions for different distribution models and I choose the bigger one which is representative of the better distribution fit. . The problem with AIC is that it tends to favor more complex models thus essentially overfits. Theta where f is a probability density or probability mass function parametrized by theta and x_1dotsx_N is your data. AIC is used to test models that are not nested but of course it can be used for neste...