## The Annals of Statistics

### A Minimax Criterion for Choosing Weight Functions for $L$-Estimates of Location

David M. Mason

#### Abstract

Let $X_1, \cdots, X_n$ be independent with common distribution $F$ symmetric about $\mu$. Let $T_n = n^{-1} \sum^n_{i = 1} J(i/(n + 1))X_{in}$ be an $L$-estimate of $\mu$ based on a weight function $J$ and the order statistics $X_{1n} \leq \cdots \leq X_{nn}$ of $X_1, \cdots, X_n$. Under very general regularity conditions $n^{1/2}T_n$ has asymptotic variance $\sigma^2(J, F)$. A weight function $J_0$ is found that minimizes the maximum of $\sigma^2(J, F)/s^2(F)$, whenever $s(F)$ is a measure of scale of a general type, as $F$ ranges over a subclass of the symmetric distributions determined by $s(F)$ and $J$ ranges over a class of weight functions also determined by $s(F)$. The sample mean and the trimmed mean arise as the solutions for particular choices of scale measures.

#### Article information

Source
Ann. Statist., Volume 11, Number 1 (1983), 317-325.

Dates
First available in Project Euclid: 12 April 2007

https://projecteuclid.org/euclid.aos/1176346082

Digital Object Identifier
doi:10.1214/aos/1176346082

Mathematical Reviews number (MathSciNet)
MR684889

Zentralblatt MATH identifier
0509.62034

JSTOR
Mason, David M. A Minimax Criterion for Choosing Weight Functions for $L$-Estimates of Location. Ann. Statist. 11 (1983), no. 1, 317--325. doi:10.1214/aos/1176346082. https://projecteuclid.org/euclid.aos/1176346082