Asymptotic and Finite-Sample Properties in Statistical Estimation
The asymptotic distribution of an estimator approximates well the central part, but less accurately the tails of its true distribution. Some properties of estimators are always non-asymptotic, regardless a widely accepted view that their properties under moderate sample sizes are inherited from the asymptotic normality. Robust estimators, advertised as resistant to heavy-tailed distributions, can be themselves heavy-tailed. They are asymptotically admissible, but not finite-sample admissible for any distribution. While the asymptotic distribution of the Newton-Raphson iteration of an estimator, starting with a consistent initial estimator, coincides with that of the non-iterated estimator, its tail-behavior is determined by that of the initial estimator. Hence, before taking a recourse to the asymptotics, we should analyze the finite-sample behavior of an estimator, whenever possible. We shall try to illustrate some distinctive differences between the asymptotic and finite-sample properties of estimators, mainly of robust ones.
Research was supported by the Grant GAČR 15-00243S. The author would like to thank the Editor for organizing the volume, and the Referee for his/her valuable comments.
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