Unbiasedness and biasedness of the Jonckheere–Terpstra and the Kruskal–Wallis tests

Finding the unbiasedness and biasedness of test statistics is important in testing a hypothesis. In this study, the unbiasedness/biasedness of the Jonckheere–Terpstra test is investigated for ordered alternatives. Our results show that the one-sided Jonckheere–Terpstra test is unbiased for the location parameter family of distributions and that the non-randomized two-sided Jonckheere–Terpstra test is biased for the shifted location parameter. Additionally, the unbiasedness/biasedness of the Kruskal–Wallis test is considered for the general two-sided alternatives. By giving a counter example, our investigation reveals that the Kruskal–Wallis test is biased against a shifted location parameter for unequal sample sizes. Our results indicate that we require to consider a bias correction for the nonparametric tests.