## The Annals of Statistics

### Minimax Variance $M$-Estimators of Location in Kolmogorov Neighbourhoods

Doug Wiens

#### Abstract

We exhibit those distributions with minimum Fisher information for location in various Kolmogorov neighbourhoods $\{F|\sup_x|F(x) - G(x)| \leq \varepsilon\}$ of a fixed, symmetric distribution $G$. The associated $M$-estimators are then most robust (in Huber's minimax sense) for location estimation within these neighbourhoods. The previously obtained solution of Huber (1964) for $G = \Phi$ and "small" $\varepsilon$ is shown to apply to all distributions with strongly unimodal densities whose score functions satisfy a further condition. The "large" $\varepsilon$ solution for $G = \Phi$ of Sacks and Ylvisaker (1972) is shown to apply under much weaker conditions. New forms of the solution are given for such distributions as "Student's" $t$, with nonmonotonic score functions. The general form of the solution is discussed.

#### Article information

Source
Ann. Statist., Volume 14, Number 2 (1986), 724-732.

Dates
First available in Project Euclid: 12 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aos/1176349949

Digital Object Identifier
doi:10.1214/aos/1176349949

Mathematical Reviews number (MathSciNet)
MR840525

Zentralblatt MATH identifier
0603.62043

JSTOR
links.jstor.org

Subjects
Primary: 62G35: Robustness
Secondary: 62G05: Estimation

#### Citation

Wiens, Doug. Minimax Variance $M$-Estimators of Location in Kolmogorov Neighbourhoods. Ann. Statist. 14 (1986), no. 2, 724--732. doi:10.1214/aos/1176349949. https://projecteuclid.org/euclid.aos/1176349949