Van der Waerden test

A non-parametric test for the homogeneity of two samples and , based on the rank statistic

where are the ranks (ordinal numbers) of the random variables in the series of joint order statistics of and ; the function is defined by the pre-selected permutation

and is the inverse function of the normal distribution with parameters . The permutation is so chosen that for a given alternative hypothesis the test will be the strongest. If , irrespective of the behaviour of and individually, the asymptotic distribution of is normal. If and are independent and normally distributed with equal variances, the test for the alternative choice or (in this case ) 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)

References

[a1]	E.L. Lehmann, "Testing statistical hypotheses" , Wiley (1986)
[a2]	M.G. Kendall, A. Stuart, "The advanced theory of statistics" , 2. Inference and relationship , Griffin (1979)