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 {X 1 , . . . , X n } a sample from F , put X n = n −1 n i=1 X i and Med n = median{X 1 , . . . , X n }. Each of X n and Med n 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 X n , the classical central limit theorem tells us: if F has finite variance σ 2
CITATION STYLE
Serfling, R. (2011). Asymptotic Relative Efficiency in Estimation. In International Encyclopedia of Statistical Science (pp. 68–72). Springer Berlin Heidelberg. https://doi.org/10.1007/978-3-642-04898-2_126
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