July 1966

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The idea of basing tests on the sample distribution function is a natural one. The Kolmogorov-Smirnov tests are of this nature. Blum, Kiefer & Rosenblatt (1961) made use of this approach to construct distribution free tests of independence. In this paper this method is further applied to the common two-sample problems of location and dispersion, the k-sample problem of location, and to the problem of dependence in bivariate samples. The rationale of the method is the following. Write the departure from the null hypothesis (against which the test must be sensitive) in terms of the true distribution functions and regard the mean value of this departure as the parameter of interest. The sample estimate of this parameter, expressed in terms of sample distribution functions, is then proposed as test statistic. This statistic is generally only dependent on the ranks of the samples and is consequently distribution free. Specifically this approach leads to Wilcoxon's two-sample test, a k-sample extension of Wilcoxon's test which is slightly different from the Kruskal-Wallis (1952) extension, the test of Ansari & Bradley (1960) for differences in dispersion, which is also a special case of a procedure proposed by Barton & David (1958), and to a test of dependence in bivariate samples which comes out to be a linear function of the rank correlation coefficients of Spearman and Kendall. The alternative k-sample extension of Wilcoxon's test has the same asymptotic relative efficiency properties as the Kruskal-Wallis test; it is however consistent against a slightly wider class than the latter. As is to be expected, the alternative test of independence behaves for large samples in much the same way as the tests of Spearman and Kendall.