ABSTRACT
This chapter discusses first-order asymptotic theory arising in statistical inference in regular parametric models when the amount of information is large, so that techniques of local linearization and use of the central limit theorem are available. In particular, at least for models with independent component random variables, the score function, being a sum of independent components, is asymptotically normal. Local linearization relates the distribution of maximum likelihood estimates to that of the score function and the unknown parameters arising in the information matrix determining the limiting distribution can be consistently estimated. A serious issue in principle concerns the asymptotic existence, uniqueness and consistency of the maximum likelihood estimate. A general result on the existence of a solution of the maximum likelihood estimating equation asymptotically close to the true parameter value is much easier to prove but is considerably less than is required.
