(k-1)-mean significance levels of nonparametric multiple comparisons procedures

We consider the nonparametric pairwise comparisons procedures derived from the Kruskal-Wallis test and from Friedman's test. For large samples the (k-1)-mean significance level is determined, i.e. the probability of concluding incorrectly that some of the first k-1 samples are unequal. We show that this probability may be larger than the simultaneous significance level \alpha. Even when the kth sample is a shift of the other k-1 samples, it may exceed \alpha, if the distributions are very skew. Here skewness is defined with Van Zwet's c-ordering of distribution functions.