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March, 1954 Universal Bounds for Mean Range and Extreme Observation
H. O. Hartley, H. A. David
Ann. Math. Statist. 25(1): 85-99 (March, 1954). DOI: 10.1214/aoms/1177728848


Consider any distribution $f(x)$ with standard deviation $\sigma$ and let $x_1, x_2 \cdots x_n$ denote the order statistics in a sample of size $n$ from $f(x).$ Further let $w_n = x_n - x_1$ denote the sample range. Universal upper and lower bounds are derived for the ratio $E(w_n)/\sigma$ for any $f(x)$ for which $a\sigma \leqq x \leqq b\sigma,$ where $a$ and $b$ are given constants. Universal upper bounds are given for $E(x_n)/\sigma$ for the case $- \infty < x < \infty.$ The upper bounds are obtained by adopting procedures of the calculus of variation on lines similar to those used by Plackett [3] and Moriguti [4]. The lower bounds are attained by singular distributions and require the use of special arguments.


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H. O. Hartley. H. A. David. "Universal Bounds for Mean Range and Extreme Observation." Ann. Math. Statist. 25 (1) 85 - 99, March, 1954.


Published: March, 1954
First available in Project Euclid: 28 April 2007

zbMATH: 0055.12801
MathSciNet: MR60775
Digital Object Identifier: 10.1214/aoms/1177728848

Rights: Copyright © 1954 Institute of Mathematical Statistics

Vol.25 • No. 1 • March, 1954
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