## The Annals of Mathematical Statistics

- Ann. Math. Statist.
- Volume 22, Number 2 (1951), 274-282.

### A Bivariate Extension of the $U$ Statistic

#### Abstract

Let $x, y$, and $z$ be three random variables with continuous cumulative distribution functions $f, g$, and $h$. In order to test the hypothesis $f = g = h$ under certain alternatives two statistics $U, V$ based on ranks are proposed. Recurrence relations are given for determining the probability of a given $(U, V)$ in a sample of $l x$'s, $m y$'s, $n z$'s and the different moments of the joint distribution of $U$ and $V$. The means, second, and fourth moments of the joint distribution are given explicitly and the limit distribution is shown to be normal. As an illustration the joint distribution of $U, V$ is given for the case $(l, m, n) = (6, 3, 3)$ together with the values obtained by using the bivariate normal approximation. Tables of the joint cumulative distribution of $U, V$ have been prepared for all cases where $l + m + n \leqq 15$.

#### Article information

**Source**

Ann. Math. Statist., Volume 22, Number 2 (1951), 274-282.

**Dates**

First available in Project Euclid: 28 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aoms/1177729647

**Digital Object Identifier**

doi:10.1214/aoms/1177729647

**Mathematical Reviews number (MathSciNet)**

MR41388

**Zentralblatt MATH identifier**

0044.14704

**JSTOR**

links.jstor.org

#### Citation

Whitney, D. R. A Bivariate Extension of the $U$ Statistic. Ann. Math. Statist. 22 (1951), no. 2, 274--282. doi:10.1214/aoms/1177729647. https://projecteuclid.org/euclid.aoms/1177729647