# Van der Waerden test

A non-parametric test for the homogeneity of two samples $Y _ {1} \dots Y _ {n}$ and $Z _ {1} \dots Z _ {m}$, based on the rank statistic

$$X = \sum _ {i = 1 } ^ { m } \Psi \left ( \frac{s ( r _ {i} ) }{m + n + 1 } \right ) ,$$

where $r _ {i}$ are the ranks (ordinal numbers) of the random variables $Z _ {i}$ in the series of joint order statistics of $Y _ {j}$ and $Z _ {i}$; the function $s( r)$ is defined by the pre-selected permutation

$$\left ( \begin{array}{c} 1 \\ s ( 1) \end{array} \begin{array}{c} \dots \\ \dots \end{array} \begin{array}{c} ( m + n) \\ s ( m + n) \end{array} \right ) ,$$

and $\Psi ( p)$ is the inverse function of the normal distribution with parameters $( 0, 1)$. The permutation is so chosen that for a given alternative hypothesis the test will be the strongest. If $m + n \rightarrow \infty$, irrespective of the behaviour of $m$ and $n$ individually, the asymptotic distribution of $X$ is normal. If $Y$ and $Z$ are independent and normally distributed with equal variances, the test for the alternative choice ${\mathsf P} ( Y < T) < {\mathsf P} ( Z < T)$ or ${\mathsf P} ( Y \langle T) \rangle {\mathsf P} ( Z < T)$( in this case $s( r) \equiv r$) is asymptotically equally as strong as the Student test. Introduced by B.L. van der Waerden [1].

#### References

 [1] B.L. van der Waerden, "Order tests for the two-sample problem and their power" Proc. Kon. Nederl. Akad. Wetensch. A , 55 (1952) pp. 453–458 [2] B.L. van der Waerden, "Mathematische Statistik" , Springer (1957) [3] L.N. Bol'shev, N.V. Smirnov, "Tables of mathematical statistics" , Libr. math. tables , 46 , Nauka (1983) (In Russian) (Processed by L.S. Bark and E.S. Kedrova)