An alternative approach for constructing small sample and limiting distributions of maximum likelihood estimators
We construct limiting and small sample distributions of maximum likelihood estimators (mle) from the property that they satisfy the first order condition (foc). The foc relates the mle of the analyzed model to the mle of an encompassing model and shows that the mle of the analyzed model is a realization from the limiting or small sample distribution of the mle of the encompassing model given that the foc holds. We can thus use the unique conditional (limiting or small sample) density of the mle of the encompassing model given that the foc holds to construct the limiting or small sample density/distribution of the mle of the analyzed model. To proof the validity of this approach and thus of the concept of an unique conditional density, we use it to construct the small sample and limiting distribution of the limited information mle and show that they are identical to resp. the sampling density and the expression discussed elsewhere in the literature. We analyze the further and relate it to existing expressions and show its limiting behavior in case of weak and strong instruments.