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March, 1983 A Minimax Criterion for Choosing Weight Functions for $L$-Estimates of Location
David M. Mason
Ann. Statist. 11(1): 317-325 (March, 1983). DOI: 10.1214/aos/1176346082


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.


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David M. Mason. "A Minimax Criterion for Choosing Weight Functions for $L$-Estimates of Location." Ann. Statist. 11 (1) 317 - 325, March, 1983.


Published: March, 1983
First available in Project Euclid: 12 April 2007

zbMATH: 0509.62034
MathSciNet: MR684889
Digital Object Identifier: 10.1214/aos/1176346082

Primary: 62G35
Secondary: 62G05

Keywords: $L$-estimates for location , minimax , order statistics , ‎weight function

Rights: Copyright © 1983 Institute of Mathematical Statistics

Vol.11 • No. 1 • March, 1983
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