## The Annals of Mathematical Statistics

### Bounds for the Variance of the Mann-Whitney Statistic

#### Abstract

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.

#### Article information

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

Dates
First available in Project Euclid: 27 April 2007

https://projecteuclid.org/euclid.aoms/1177706794

Digital Object Identifier
doi:10.1214/aoms/1177706794

Mathematical Reviews number (MathSciNet)
MR93875

Zentralblatt MATH identifier
0081.14002

JSTOR