Asymptotic relative efficiency of estimators

For statistical estimation problems, it is typical and even desirable that more than one reasonable estimator can arise for consideration. One natural and time-honored approach for choosing an estimator is simply to compare the sample sizes at which the competing estimators meet a given standard of performance. This depends upon the chosen measure of performance and upon the particular population distribution F . For example, we might compare the sample mean versus the sample median for location estimation. Consider a distribution function F with density function f symmetric about an unknown point θ to be estimated. For {X1, . . . , Xn} a sample from F , putXn = n−1 ∑n i=1 Xi and Medn = median{X1, . . . , Xn}. Each of Xn and Medn is a consistent estimator of θ in the sense of convergence in probability to θ as the sample size n → ∞. To choose between these estimators we need to use further information about their performance. In this regard, one key aspect is efficiency, which answers: