## The Annals of Statistics

### Tail-Behavior of Location Estimators

Jana Jureckova

#### Abstract

Let $X_1, \cdots, X_n$ be a sample from a population with density $f(x - \theta)$ such that $f$ is symmetric and positive. It is proved that the tails of the distribution of a translation-invariant estimator of $\theta$ tend to 0 at most $n$ times faster than the tails of the basic distribution. The sample mean is shown to be good in this sense for exponentially-tailed distributions while it becomes poor if there is contamination by a heavy-tailed distribution. The rates of convergence of the tails of robust estimators are shown to be bounded away from the lower as well as from the upper bound.

#### Article information

Source
Ann. Statist., Volume 9, Number 3 (1981), 578-585.

Dates
First available in Project Euclid: 12 April 2007

https://projecteuclid.org/euclid.aos/1176345461

Digital Object Identifier
doi:10.1214/aos/1176345461

Mathematical Reviews number (MathSciNet)
MR615433

Zentralblatt MATH identifier
0476.62032

JSTOR