Open Access
September, 1975 Descriptive Statistics for Nonparametric Models II. Location
P. J. Bickel, E. L. Lehmann
Ann. Statist. 3(5): 1045-1069 (September, 1975). DOI: 10.1214/aos/1176343240


Measures of location (without assumption of symmetry) are defined as functionals satisfying certain equivariance and order conditions. Three classes of such measures are discussed whose estimators are respectively linear functions of order statistics, $R$-estimators and $M$-estimators. It is argued that such measures can be compared in terms of the (asymptotic) efficiencies of their estimators. Of the three classes considered, it is found that trimmed expectations (and certain other weighted quantiles) are the only ones which are both robust and whose estimators have guaranteed high efficiency relative to the mean $\bar{X}$ for all underlying distributions.


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P. J. Bickel. E. L. Lehmann. "Descriptive Statistics for Nonparametric Models II. Location." Ann. Statist. 3 (5) 1045 - 1069, September, 1975.


Published: September, 1975
First available in Project Euclid: 12 April 2007

zbMATH: 0321.62055
MathSciNet: MR395021
Digital Object Identifier: 10.1214/aos/1176343240

Primary: 62G99
Secondary: 62G05 , 62G20 , 62G35

Keywords: efficiency , estimators derived from rank tests , heavy-tailed distributions , Huber's maximum likelihood estimators , linear combinations of order statistics , measures of location , robustness , Trimmed means , Tukey model

Rights: Copyright © 1975 Institute of Mathematical Statistics

Vol.3 • No. 5 • September, 1975
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