## The Annals of Mathematical Statistics

### On a Test of Whether one of Two Random Variables is Stochastically Larger than the Other

#### 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 $x$.

#### Article information

Source
Ann. Math. Statist. Volume 18, Number 1 (1947), 50-60.

Dates
First available in Project Euclid: 28 April 2007

http://projecteuclid.org/euclid.aoms/1177730491

Digital Object Identifier
doi:10.1214/aoms/1177730491

Mathematical Reviews number (MathSciNet)
MR22058

Zentralblatt MATH identifier
0041.26103

JSTOR