The Rank Version of von Neumann's Ratio Test for Randomness

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Abstract
Although rank tests for randomness were proposed in the literature as early as 1943, no such test has gained wide acceptance comparable to, say, Spearman's rho test. This may be due to the lack of small-sample theory and of tables of critical values to enable such a test to be carried out on small samples. In this article, we consider the rank version of the von Neumann ratio statistic and we obtain the critical values of this statistic under the randomization hypothesis. In a Monte Carlo experiment we then show that the resulting nonparametric test for randomness has far greater power than the test based on the number of runs up and down. Moreover, under normality, its power vis-a-vis the normal theory von Neumann ratio test is also very good. It is therefore suggested that with the tables presented in this article, the rank von Neumann ratio test for randomness provides an easy and powerful alternative to nonparametric tests now in common use.

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