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Universal Bounds for Mean Range and Extreme Observation
H. O. Hartley and H. A. David
The Annals of Mathematical Statistics
Vol. 25, No. 1 (Mar., 1954), pp. 85-99
Published by: Institute of Mathematical Statistics
Stable URL: http://www.jstor.org/stable/2236514
Page Count: 15
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Consider any distribution f(x) with standard deviation σ and let x1, x2 ⋯ xn denote the order statistics in a sample of size n from f(x). Further let wn = xn - x1 denote the sample range. Universal upper and lower bounds are derived for the ratio E(wn)/σ for any f(x) for which aσ ≤ x ≤ bσ, where a and b are given constants. Universal upper bounds are given for E(xn)/σ 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  and Moriguti . The lower bounds are attained by singular distributions and require the use of special arguments.
The Annals of Mathematical Statistics © 1954 Institute of Mathematical Statistics