Abstract
A class of robust location estimators called weighted randomly trimmed means are introduced and not only their consistency and asymptotic normality are proved, but their influence functions, asymptotic variances and breakdown points are also derived. They possess the same breakdown points as the median, and some of them own higher asymptotic relative efficiencies at the heavy-tailed distributions than some other well-known location estimators; whereas the trimmed means, Winsorized means and Huber’s M-estimator possess higher asymptotic relative efficiencies at the light-tailed distributions, in which Huber’s M-estimator is the most robust.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
J. W. Tukey, A survey of sampling from contaminated distributions, in Contributions to Probability and Statistics (ed. by I. Olkin et al.), Stanford University Press, Palo Alto, Calif., 1960, 448–485.
J. W. Tukey D. H. Mclaughlin (1963) ArticleTitleLess vulnerable confidence and significance procedures for location based on a single sample: Trimming/Winsorization I Sankhyā A 25 331–352
P. J. Huber (1964) ArticleTitleRobust estimation of a location parameter Ann. Math. Statist. 35 73–101
R. Y. Liu (1990) ArticleTitleOn a notion of data depth based on random simplices Ann. Statist. 18 405–414
Y. Zuo H. Cui X. He (2004) ArticleTitleOn the Stahel–Donoho estimator and depth-weighted means for multivariate data Ann. Statist. 32 169–190
Y. Zuo, Multivariate trimmed means based on data depth, in Statistical Data Analysis Based on the L1-Norm and Related Methods (ed. by Y. Dodge), 2002, 313–322.
Y. Zuo, Trimming with a random fraction of trimmed points, A report at the Joint Meeting of CSPS/IMS, 2005.
H. Cui Y. Tian (1994) ArticleTitleEstimation of the projection absolute median deviation and its application (in chinese) J. Systems Sci. Math. Sci. 14 63–72
F. R. Hampel, Contributions to the theory of robust estimation, PhD thesis, University of California, Berkeley, 1968.
F. R. Hampel (1974) ArticleTitleThe influence curve and its role in robust estimation J. Am. Statist. Assoc. 69 383–393 Occurrence Handle10.2307/2285666
J. A. Reeds (1976) On the definition of von Mises functionals, Research Report S 44 Department of Statistics, Harvard University Cambridge, Mass
D. D. Boos R. J. Serfling (1980) ArticleTitleA note on differentials and the CLT ans LIL for statistical functions, with application to M-estimates Ann. Statist. 8 618–624
P. J. Huber (1981) Robust Statistics Wiley New York Occurrence Handle10.1002/0471725250
D. C. Hoaglin, F. Mosteller, and J. W. Tukey, Understanding Robust and Exploratory Data Analysis, John Wiley & Sons Inc., 1983.
D. L. Donoho and P. J. Huber, The notion of breakdown point, in A Festschrift foe Erich L. Lehmann (ed. by P. J. Bickel, K. A. Doksum, and J. L. Hodges), Wadsworth, Belmont, CA, 1983, 157–184.
D. S. Moore (1968) ArticleTitleAn elementary proof of asymptotic normality of linear functions of order statistics Annals of Mathematical Statistics 39 263–265
X. Chen G. Chai (1993) Nonparametric Statistics (in Chinese) East China Normal University Press Shanghai
F. R. Hampel E. M. Ronchetti P. J. Rousseeuw W. A. Stahel (1986) Robust Statistics: The Approach Based on Influence Functions Wiley New York
Author information
Authors and Affiliations
Corresponding author
Additional information
This research is supported by the National Natural Science Foundation of China (Grant No. 10371012, 10231030, and 40574020).
Rights and permissions
About this article
Cite this article
Wang, T., Li, Y. & Cui, H. On Weighted Randomly Trimmed Means. Jrl Syst Sci & Complex 20, 47–65 (2007). https://doi.org/10.1007/s11424-007-9004-7
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/s11424-007-9004-7