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June, 1965 Some Bounds for Expected Values of Order Statistics
Mir M. Ali, Lai K. Chan
Ann. Math. Statist. 36(3): 1055-1057 (June, 1965). DOI: 10.1214/aoms/1177700081

Abstract

Let the function $F(x)$ be a distribution function for a continuous symmetric distribution, and let $X_{(i)}$ represent the $i$th order statistics from a sample of size $n$. It is shown in this paper that for $i \geqq (n + 1)/2$ $E(X_{(i)}) \geqq G(i/(n + 1)) \quad\text{if} F \text{is unimodal}$ and $E(X_{(i)}) \leqq G(i/(n + 1)) \quad\text{if} F is U\text{shaped},$ where $x = G(u)$ is the inverse function of $F(x) = u$. The definitions of unimodal and $U$-shaped distributions are given in Section 3. The above inequalities are of interest, since it is known (Blom (1958), Chapters 5 and 6) that for sufficiently large $n$ the bound $G(i/(n + 1))$ approaches $E(X_{(i)})$.

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Mir M. Ali. Lai K. Chan. "Some Bounds for Expected Values of Order Statistics." Ann. Math. Statist. 36 (3) 1055 - 1057, June, 1965. https://doi.org/10.1214/aoms/1177700081

Information

Published: June, 1965
First available in Project Euclid: 27 April 2007

zbMATH: 0136.40907
MathSciNet: MR177482
Digital Object Identifier: 10.1214/aoms/1177700081

Rights: Copyright © 1965 Institute of Mathematical Statistics

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Vol.36 • No. 3 • June, 1965
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