March, 1954 Universal Bounds for Mean Range and Extreme Observation

H. O. Hartley, H. A. David

Abstract

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.

Citation

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

Information

JOURNAL ARTICLE
15 PAGES

Vol.25 • No. 1 • March, 1954