# Asymptotic and Finite-Sample Properties in Statistical Estimation

Chapter
Part of the Fields Institute Communications book series (FIC, volume 76)

## Abstract

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.

## References

1. 1.
Field, C.A., Ronchetti, E.: Small Sample Asymptotics. Lecture Notes–Monograph Series. Institute of Mathematical Statistics, Hayward (1990)
2. 2.
He, X., Jurečková, J., Koenker, R., Portnoy, S.: Tail behavior of regression estimators and their breakdown points. Econometrica 58, 1195–1214 (1990)
3. 3.
Jurečková, J.: Tail behavior of location estimators. Ann. Stat. 9, 578–585 (1981)
4. 4.
Jurečková, J.: Finite sample distribution of regression quantiles. Stat. Probab. Lett. 80, 1940–1946 (2010)
5. 5.
Jurečková, J.: Tail-behavior of estimators and of their one-step versions. Journal de la Société Francaise de Statistique 153/1, 44–51 (2012)Google Scholar
6. 6.
Jurečková, J., Klebanov, L.B.: Inadmissibility of robust estimators with respect to L1 norm. In: Dodge, Y. (ed.) L 1-Statistical Procedures and Related Topics. Lecture Notes–Monographs Series, vol. 31, pp. 71–78. Institute of Mathematical Statistics, Hayward (1997)Google Scholar
7. 7.
Jurečková, J., Klebanov, L.B.: Trimmed, Bayesian and admissible estimators. Stat. Probab. Lett. 42, 47–51 (1998)
8. 8.
Jurečková, J., Picek, J.: Finite-sample behavior of robust estimators. In: Chen, S., Mastorakis, N., Rivas-Echeverria, F., Mladenov, V. (eds.) Recent Researches in Instrumentation, Measurement, Circuits and Systems pp. 15–20. ISBN: 978-960-474-282-0. ISSN: 1792-8575Google Scholar
9. 9.
Jurečková, J., Portnoy, S.: Asymptotics for one-step M-estimators in regression with application to combining efficiency and high breakdown point. Commun. Stat. A 16, 2187–2199 (1987)
10. 10.
Jurečková, J., Sabolová, R.: Finite-sample density and its small sample asymptotic approximation. Stat. Probab. Lett. 81, 1311–1318 (2011)
11. 11.
Jurečková, J., Koenker, R., Portnoy, S.: Tail behavior of the least-squares estimator. Stat. Probab. Lett. 55, 377–384 (2001)
12. 12.
Mizera, I., Mueller, C.H.: Breakdown points and variation exponents of robust M-estimators in linear models. Ann. Stat. 27, 1164–1177 (1999)
13. 13.
Portnoy, S., Jurečková, J.: On extreme regression quantiles. Extremes 2(3), 227–243 (1999)
14. 14.
Sabolová, R.: Small sample inference for regression quantiles. KPMS Preprint 70, Charles University in Prague (2012)Google Scholar
15. 15.
Zuo, Y.: Finite sample tail behavior of the multivariate trimmed mean based on Tukey-Donoho halfspace depth. Metrika 52, 69–75 (2000)
16. 16.
Zuo, Y.: Finite sample tail behavior of Hodges-Lehmann type estimators. Statistics 35, 557–568 (2001)
17. 17.
Zuo, Y.: Finite sample tail behavior of multivariate location estimators. J. Multivar. Anal. 85, 91–105 (2003) 