Maximum likelihood estimation of different models (which all model the same variable and assume the same likelihood function) is done by a different method for each model. Simple numerical maximization of the likelihood is not possible due to the "curse of dimensionality" and consequently methods like
- composite likelihood estimation
- estimating a subset of parameters by a moment estimator (and the rest by ML)
- iteratively estimating subsets of parameters
are employed. Assuming all estimation procedures yield consistent parameter estimates but some are more efficient than others, is it sensible to compare the models via AIC or BIC? What if some procedures are inconsistent?