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June, 1951 A Bivariate Extension of the $U$ Statistic
D. R. Whitney
Ann. Math. Statist. 22(2): 274-282 (June, 1951). DOI: 10.1214/aoms/1177729647


Let $x, y$, and $z$ be three random variables with continuous cumulative distribution functions $f, g$, and $h$. In order to test the hypothesis $f = g = h$ under certain alternatives two statistics $U, V$ based on ranks are proposed. Recurrence relations are given for determining the probability of a given $(U, V)$ in a sample of $l x$'s, $m y$'s, $n z$'s and the different moments of the joint distribution of $U$ and $V$. The means, second, and fourth moments of the joint distribution are given explicitly and the limit distribution is shown to be normal. As an illustration the joint distribution of $U, V$ is given for the case $(l, m, n) = (6, 3, 3)$ together with the values obtained by using the bivariate normal approximation. Tables of the joint cumulative distribution of $U, V$ have been prepared for all cases where $l + m + n \leqq 15$.


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D. R. Whitney. "A Bivariate Extension of the $U$ Statistic." Ann. Math. Statist. 22 (2) 274 - 282, June, 1951.


Published: June, 1951
First available in Project Euclid: 28 April 2007

zbMATH: 0044.14704
MathSciNet: MR41388
Digital Object Identifier: 10.1214/aoms/1177729647

Rights: Copyright © 1951 Institute of Mathematical Statistics

Vol.22 • No. 2 • June, 1951
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