Open Access
December, 1957 Bounds for the Variance of the Mann-Whitney Statistic
Z. W. Birnbaum, Orval M. Klose
Ann. Math. Statist. 28(4): 933-945 (December, 1957). DOI: 10.1214/aoms/1177706794


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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Z. W. Birnbaum. Orval M. Klose. "Bounds for the Variance of the Mann-Whitney Statistic." Ann. Math. Statist. 28 (4) 933 - 945, December, 1957.


Published: December, 1957
First available in Project Euclid: 27 April 2007

zbMATH: 0081.14002
MathSciNet: MR93875
Digital Object Identifier: 10.1214/aoms/1177706794

Rights: Copyright © 1957 Institute of Mathematical Statistics

Vol.28 • No. 4 • December, 1957
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