## Annals of Mathematical Statistics

### Some Order Statistic Distributions for Samples of Size Four

John E. Walsh

#### Abstract

Let $x_1, x_2, x_3, x_4$ represent the values of a sample of size four drawn from a normal population. There is no loss of generality in assuming that the distribution function of this population has zero mean and unit vriance. Denote it by $N(0, 1)$. Let $x_(i)$ be the ith largest of $x_1, x_2, x_3, x_4$. The purpose of this note is to determine the joint distribution of $x{(4)} + x_{(3)} - x_{(2)} - x_{(1)}, x_{(4)} - x_{(3)} + x_{(2)} -x_{(1)}$, and $x_{(4)} - x_{(3)} - x_{(2)} + x_{(1)}$, and derive from this joint distribution the joint distributions of these statistics taken in pairs, also the distribution of each statistic itself.

#### Article information

Source
Ann. Math. Statist., Volume 17, Number 2 (1946), 246-248.

Dates
First available in Project Euclid: 28 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aoms/1177730988

Digital Object Identifier
doi:10.1214/aoms/1177730988

Mathematical Reviews number (MathSciNet)
MR16616

Zentralblatt MATH identifier
0060.30014

JSTOR