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 X for all underlying distributions.
Received April 1973; revised September 1974.
This research was partially supported by the Office of Naval Research, Contract NONR N00014-69-A-0200-1038.
Work done while the author was Research Professor for the Adolph C. and Mary Sprague Miller Institute for Basic Research in Science, University of California, Berkeley.
AMS 1970 subject classifications. Primary 62G99; Secondary 62G05, 62G20, 62G35.
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Bickel, P.J., Lehmann, E.L. (2012). Descriptive Statistics for Nonparametric Models II. Location. In: Rojo, J. (eds) Selected Works of E. L. Lehmann. Selected Works in Probability and Statistics. Springer, Boston, MA. https://doi.org/10.1007/978-1-4614-1412-4_43
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