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.
Citation
John E. Walsh. "Some Order Statistic Distributions for Samples of Size Four." Ann. Math. Statist. 17 (2) 246 - 248, June, 1946. https://doi.org/10.1214/aoms/1177730988
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