A test for the maximum of a large number of values of Kendall's τ is obtained by an application of a Poisson limit theorem for symmetric statistics proved by Silverman & Brown (1978). The test depends on the number of comparisons rather than the sample size being large. It is shown that the asymptotics are always practically useful. A table of critical values for the test statistic is given, together with an illustrative application.
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Best et al. (1983) studied this question.