Annals of Mathematical Statistics

Some Order Statistic Distributions for Samples of Size Four

John E. Walsh

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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.

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

First available in Project Euclid: 28 April 2007

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Walsh, John E. Some Order Statistic Distributions for Samples of Size Four. Ann. Math. Statist. 17 (1946), no. 2, 246--248. doi:10.1214/aoms/1177730988.

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