Asymptotic Relative Efficiency in Estimation

For statistical estimation problems, it is typical and even desirable that several reasonable estimators can arise for consideration. For example, the mean and median parameters of a symmetric distribution coincide, and so the sample mean and the sample median become competing estimators of the point of symmetry. Which is preferred? By what criteria shall we make a choice? One natural and time-honored approach is simply to compare the sample sizes at which two competing estimators meet a given standard of performance. This depends upon the chosen measure of performance and upon the particular population distribution F . To make the discussion of sample mean versus sample median more precise, 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 , put Xn = 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: How spread out about θ is the sampling distribution of the estimator? The smaller the variance in its sampling distribution, the more “efficient” is that estimator. Here we consider “large-sample” sampling distributions. For Xn, the classical central limit theorem tells us: if F has finite variance σ F , then the sampling distribution of Xn is approximately N(θ, σ F/n), i.e., Normal with mean θ and variance σ 2 F/n. For Medn, a similar