## The Annals of Statistics

### Robust Location Estimates

Rudolf Beran

#### Abstract

Measures of location differentiable at every density in the Hellinger metric are constructed in this paper. Differentiability entitles these location functionals to the label "robust," even though their influence curves need not be bounded and continuous. The latter properties are, in fact, associated with functionals differentiable in the Prokhorov metric. A Hellinger metric concept of minimax robustness of a location measure at a density shape $f$ is developed. Asymptotically optimal estimators are found for minimax robust location measures. Since, at $f$, their asymptotic variance equals the reciprocal of Fisher information, asymptotic efficiency at $f$ and robustness near $f$ prove compatible.

#### Article information

Source
Ann. Statist. Volume 5, Number 3 (1977), 431-444.

Dates
First available in Project Euclid: 12 April 2007

http://projecteuclid.org/euclid.aos/1176343841

JSTOR

Digital Object Identifier
doi:10.1214/aos/1176343841

Mathematical Reviews number (MathSciNet)
MR448699

Zentralblatt MATH identifier
0381.62032

Subjects
Primary: 62G35: Robustness
Secondary: 62E20: Asymptotic distribution theory

#### Citation

Beran, Rudolf. Robust Location Estimates. Ann. Statist. 5 (1977), no. 3, 431--444. doi:10.1214/aos/1176343841. http://projecteuclid.org/euclid.aos/1176343841.