## Access

You are not currently logged in.

Access your personal account or get JSTOR access through your library or other institution:

## If You Use a Screen Reader

This content is available through Read Online (Free) program, which relies on page scans. Since scans are not currently available to screen readers, please contact JSTOR User Support for access. We'll provide a PDF copy for your screen reader.

# 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
Stable URL: http://www.jstor.org/stable/2236514
Page Count: 15
Preview not available

## Abstract

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 [3] and Moriguti [4]. The lower bounds are attained by singular distributions and require the use of special arguments.

• 85
• 86
• 87
• 88
• 89
• 90
• 91
• 92
• 93
• 94
• 95
• 96
• 97
• 98
• 99