## The Annals of Mathematical Statistics

- Ann. Math. Statist.
- Volume 25, Number 1 (1954), 85-99.

### Universal Bounds for Mean Range and Extreme Observation

#### 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.

#### Article information

**Source**

Ann. Math. Statist., Volume 25, Number 1 (1954), 85-99.

**Dates**

First available in Project Euclid: 28 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aoms/1177728848

**Digital Object Identifier**

doi:10.1214/aoms/1177728848

**Mathematical Reviews number (MathSciNet)**

MR60775

**Zentralblatt MATH identifier**

0055.12801

**JSTOR**

links.jstor.org

#### Citation

Hartley, H. O.; David, H. A. Universal Bounds for Mean Range and Extreme Observation. Ann. Math. Statist. 25 (1954), no. 1, 85--99. doi:10.1214/aoms/1177728848. https://projecteuclid.org/euclid.aoms/1177728848