## The Annals of Statistics

### Descriptive Statistics for Nonparametric Models II. Location

#### 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

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