Asymptotic (Relative) Efficiency of Estimators

This chapter deals with the quality of estimators of unknown parameters in the presence of large sample sizes. It starts with a proof of the famous information inequality of Fréchet-Cramér-Rao that, under certain regularity conditions, provides a lower bound for the variance of an estimator. After pointing out the bias-variance-tradeoff in connection with minimizing the mean squared estimation error, a proof of the multivariate information inequality is given. Under certain regularity conditions, this inequality provides a lower bound for the covariance matrix of an estimator with respect to the Lwner half-order for matrices. After introducing the concept of a best asymptotically normal (BAN) estimator and stating the famous LeCam-Bahadur theorem, the chapter proceeds with the method of moments, which is another important principle of constructing estimators. Other topics include the construction of BAN estimators and asymptotic relative Pitman efficiency. The chapter concludes with an obvious problem, namely how to estimate the center of a symmetric distribution