The Annals of Mathematical Statistics

Bounds for the Variance of the Mann-Whitney Statistic

Z. W. Birnbaum and Orval M. Klose

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Let $X, Y$ be independent random variables with continuous cumulative probability functions and let $$p = \mathrm{Pr}\{Y < X\}.$$ For the variance of the Mann-Whitney statistic $U,$ upper and lower bounds are obtained in terms of $p$, for the case of any $X$ and $Y$ as well as for the case of stochastically comparable $X, Y$. The results for the case of stochastic comparability are new, while the inequalities in the case of arbitrary $X, Y$ have either been obtained by van Dantzig or are a consequence of other inequalities due to van Dantzig.

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Ann. Math. Statist., Volume 28, Number 4 (1957), 933-945.

First available in Project Euclid: 27 April 2007

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Birnbaum, Z. W.; Klose, Orval M. Bounds for the Variance of the Mann-Whitney Statistic. Ann. Math. Statist. 28 (1957), no. 4, 933--945. doi:10.1214/aoms/1177706794.

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