The Stein Phenomenon and Superefficiency

This lecture delves into the Stein Phenomenon, where the James-Stein Estimator dominates the Maximum Likelihood Estimator (MLE) in Gaussian estimation under quadratic loss. It challenges the belief in the MLE's universal optimality, showcasing the benefits of bias in high-dimensional statistics. The lecture also covers Hodges' Superefficient Estimator, highlighting its superiority over the MLE in certain scenarios. The discussion extends to asymptotic optimality, asymptotically Gaussian estimators, and the Cramér-Rao bound. Through examples and proofs, the lecture explores the intricacies of estimation theory, emphasizing the importance of regular sequences of estimators and the implications of superefficiency.