The Annals of Mathematical Statistics

A Bivariate Extension of the $U$ Statistic

D. R. Whitney

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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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Ann. Math. Statist., Volume 22, Number 2 (1951), 274-282.

First available in Project Euclid: 28 April 2007

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Whitney, D. R. A Bivariate Extension of the $U$ Statistic. Ann. Math. Statist. 22 (1951), no. 2, 274--282. doi:10.1214/aoms/1177729647.

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