Distributions of Maximum Likelihood Estimators and Model Comparisons

Experimental data need to be assessed for purposes of model identification, estimation of model parameters and consequences of misspecified model fits. Here the first and third factors are considered via analytic formulations for the distribution of the maximum likelihood estimates. When estimating this distribution with statistics, it is a tradition to invert the roles of population quantities and quantities that have been estimated from the observed sample. If the model is known, simulations, normal approximations and p*-formula methods can be used. However, exact analytic methods for describing the estimator density are recommended. One of the methods (TED) can be used when the data generating model differs from the estimation model, which allows for the estimation of common parameters across a suite of candidate models. Information criteria such as AIC can be used to pick a winning model. AIC is however approximate and generally only asymptotically correct. For fairly simple models, where expressions remain tractable, the exact estimator density under TED allows for comparisons between models. This is illustrated via a novel information criterion. Three linear models are compared and fitted to econometric data on patent filings.