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July, 1974 Linear Functions of Order Statistics with Smooth Weight Functions
Stephen M. Stigler
Ann. Statist. 2(4): 676-693 (July, 1974). DOI: 10.1214/aos/1176342756

Abstract

This paper considers linear functions of order statistics of the form $S_n = n^{-1} \sum J(i/(n + 1))X_{(i)}$. The main results are that $S_n$ is asymptotically normal if the second moment of the population is finite and $J$ is bounded and continuous a.e. $F^{-1}$, and that this first result continues to hold even if the unordered observations are not identically distributed. The moment condition can be discarded if $J$ trims the extremes. In addition, asymptotic formulas for the mean and variance of $S_n$ are given for both the identically and non-identically distributed cases. All of the theorems of this paper apply to discrete populations, continuous populations, and grouped data, and the conditions on $J$ are easily checked (and are satisfied by most robust statistics of the form $S_n$). Finally, a number of applications are given, including the trimmed mean and Gini's mean difference, and an example is presented which shows that $S_n$ may not be asymptotically normal if $J$ is discontinuous.

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Stephen M. Stigler. "Linear Functions of Order Statistics with Smooth Weight Functions." Ann. Statist. 2 (4) 676 - 693, July, 1974. https://doi.org/10.1214/aos/1176342756

Information

Published: July, 1974
First available in Project Euclid: 12 April 2007

zbMATH: 0286.62028
MathSciNet: MR373152
Digital Object Identifier: 10.1214/aos/1176342756

Subjects:
Primary: 62G30
Secondary: 60F05 , 62E20 , 62G35

Keywords: moments of order statistics , order statistics , robust estimation , Trimmed means

Rights: Copyright © 1974 Institute of Mathematical Statistics

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Vol.2 • No. 4 • July, 1974
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