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].

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