## The Annals of Statistics

### Linear Functions of Order Statistics with Smooth Weight Functions

Stephen M. Stigler

#### 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.

#### Article information

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

Dates
First available in Project Euclid: 12 April 2007

https://projecteuclid.org/euclid.aos/1176342756

Digital Object Identifier
doi:10.1214/aos/1176342756

Mathematical Reviews number (MathSciNet)
MR373152

Zentralblatt MATH identifier
0286.62028

JSTOR