Abstract
Let $x$ and $y$ be two random variables with continuous cumulative distribution functions $f$ and $g$. A statistic $U$ depending on the relative ranks of the $x$'s and $y$'s is proposed for testing the hypothesis $f = g$. Wilcoxon proposed an equivalent test in the Biometrics Bulletin, December, 1945, but gave only a few points of the distribution of his statistic. Under the hypothesis $f = g$ the probability of obtaining a given $U$ in a sample of $n x's$ and $m y's$ is the solution of a certain recurrence relation involving $n$ and $m$. Using this recurrence relation tables have been computed giving the probability of $U$ for samples up to $n = m = 8$. At this point the distribution is almost normal. From the recurrence relation explicit expressions for the mean, variance, and fourth moment are obtained. The 2rth moment is shown to have a certain form which enabled us to prove that the limit distribution is normal if $m, n$ go to infinity in any arbitrary manner. The test is shown to be consistent with respect to the class of alternatives $f(x) > g(x)$ for every $x$.
Keywords
MathematicsCombinatoricsMoment (physics)Random variableStatisticStatisticsDistribution (mathematics)InfinityNormal distributionTest statisticCumulative distribution functionLimit (mathematics)Statistical hypothesis testingMathematical analysisProbability density functionPhysics
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Publication Info
- Year
- 1947
- Type
- article
- Volume
- 18
- Issue
- 1
- Pages
- 50-60
- Citations
- 13129
- Access
- Closed
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Cite This
H. B. Mann,
Douglas R. Whitney
(1947).
On a Test of Whether one of Two Random Variables is Stochastically Larger than the Other.
The Annals of Mathematical Statistics
, 18
(1)
, 50-60.
https://doi.org/10.1214/aoms/1177730491
Identifiers
- DOI
- 10.1214/aoms/1177730491