The Annals of Statistics

Linear Functions of Order Statistics with Smooth Weight Functions

Stephen M. Stigler

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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.

Article information

Ann. Statist., Volume 2, Number 4 (1974), 676-693.

First available in Project Euclid: 12 April 2007

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Zentralblatt MATH identifier


Primary: 62G30: Order statistics; empirical distribution functions
Secondary: 60F05: Central limit and other weak theorems 62E20: Asymptotic distribution theory 62G35: Robustness

Order statistics robust estimation moments of order statistics trimmed means


Stigler, Stephen M. Linear Functions of Order Statistics with Smooth Weight Functions. Ann. Statist. 2 (1974), no. 4, 676--693. doi:10.1214/aos/1176342756.

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  • See Correction: Stephen M. Stigler. Note: Correction to Linear Functions of Order Statistics with Smooth Weight Functions. Ann. Statist., Volume 7, Number 2 (1979), 466--466.