The Annals of Statistics

Descriptive Statistics for Nonparametric Models II. Location

P. J. Bickel and E. L. Lehmann

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Abstract

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.

Article information

Source
Ann. Statist., Volume 3, Number 5 (1975), 1045-1069.

Dates
First available in Project Euclid: 12 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aos/1176343240

Digital Object Identifier
doi:10.1214/aos/1176343240

Mathematical Reviews number (MathSciNet)
MR395021

Zentralblatt MATH identifier
0321.62055

JSTOR
links.jstor.org

Subjects
Primary: 62G99: None of the above, but in this section
Secondary: 62G05: Estimation 62G20: Asymptotic properties 62G35: Robustness

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

Citation

Bickel, P. J.; Lehmann, E. L. Descriptive Statistics for Nonparametric Models II. Location. Ann. Statist. 3 (1975), no. 5, 1045--1069. doi:10.1214/aos/1176343240. https://projecteuclid.org/euclid.aos/1176343240


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See also

  • Part I: P. J. Bickel, E. L. Lehmann. Descriptive Statistics for Nonparametric Models I. Introduction. Ann. Statist., Volume 3, Number 5 (1975), 1038--1044.
  • Part III: P. J. Bickel, E. L. Lehmann. Descriptive Statistics for Nonparametric Models. III. Dispersion. Ann. Statist., Volume 4, Number 6 (1976), 1139--1158.