Consistent estimators of the asymptotic covariance matrix of vectors of U-statistics are used in constructing asymptotic confidence regions for vectors of Kendall's correlation coefficients corresponding to various pairs of components of a random vector. The regions are products of intervals computed by means of a critical value from multivariate normal distribution. The regularity of the asymptotic covariance matrix of the vector of Kendall's sample coefficients is proved in the case of sampling from continuous multivariate distribution under mild conditions. The results are applied also to confidence intervals for the coefficient of agreement. The coverage and length of the obtained (multivariate) product of intervals are illustrated by simulation.
The testing of the null hypothesis of no treatment effect against the alternative of increasing treatment effect by means of rank statistics is extended from the classical Friedman random blocks model into an unbalanced design allowing treatments not to be applied simultaneously in each random block. The asymptotic normality of the constructed rank test statistic is proved both in the setting not allowing ties and also for models with presence of ties. As a by-product of the proofs a multiple comparisons rule based on rank statistics is obtained for the case when the null hypothesis of no treatment effect is tested against the general alternative of its negation.
Explicit formulas for the non-centrality parameters of the limiting chi-square distribution of proposed multisample rank based test statistics, aimed at testing the hypothesis of the simultaneous equality of location and scale parameters of underlying populations, are obtained by means of a general assertion concerning the location-scale test statistics. The finite sample behaviour of the proposed tests is discussed and illustrated by simulation estimates of the rejection probabilities. A modification for ties of a class of multisample location and scale test statistics, based on ranks and including the proposed test statistics, is presented. It is shown that under the validity of the null hypothesis these modified test statistics are asymptotically chi-square distributed provided that the score generating functions fulfill the imposed regularity conditions. An essential assumption is that the matrix, appearing in these conditions, is regular. Conditions sufficient for the validity of this assumption are also included.