In a rough sense, Theorem 9.14 shows that the maximum likelihood estimator achieves the Cramér–Rao lower bound asymptotically, which suggests that it is asymptotically fully efficient. In this chapter we explore results on asymptotic optimality formalizing notions of asymptotic efficiency and showing that maximum likelihood or similar estimators are efficient in regular cases. Notions of asymptotic efficiency are quite technical and involved, and the treatment here is limited. Our main goal is to derive a result from Hájek (1972), Theorem 16.25 below, which shows that the maximum likelihood estimator is locally asymptotically minimax.
KeywordsPosterior Distribution Marginal Distribution Maximum Likelihood Estimator Fisher Information Error Loss
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