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March, 1953 On the Distribution of the Expected Values of the Order Statistics
Wassily Hoeffding
Ann. Math. Statist. 24(1): 93-100 (March, 1953). DOI: 10.1214/aoms/1177729086


Let $X_1, X_2, \cdots, X_n$ be independent with a common distribution function $F(x)$ which has a finite mean, and let $Z_{n1} \leqq Z_{n2} \leqq \cdots \leqq Z_{nn}$ be the ordered values $X_1, \cdots, X_n$. The distribution of the $n$ values $EZ_{n1}, \cdots, EZ_{nn}$ on the real line is studied for large $n$. In particular, it is shown that as $n \rightarrow \infty$, the corresponding distribution function converges to $F(x)$ and any moment of that distribution converges to the corresponding moment of $F(x)$ if the latter exists. The distribution of the values $Ef(Z_{nm})$ for certain functions $f(x)$ is also considered.


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Wassily Hoeffding. "On the Distribution of the Expected Values of the Order Statistics." Ann. Math. Statist. 24 (1) 93 - 100, March, 1953.


Published: March, 1953
First available in Project Euclid: 28 April 2007

zbMATH: 0050.13603
MathSciNet: MR54197
Digital Object Identifier: 10.1214/aoms/1177729086

Rights: Copyright © 1953 Institute of Mathematical Statistics


Vol.24 • No. 1 • March, 1953
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