Bias of Nonparametric Goodness-of-Fit Tests Relative to Certain Pairs of Competing Hypotheses

The application of the nonparametric Anderson–Darling, Cramer–Mises–Smirnov, Kuiper, Watson, Kolmogorov, and Zhang goodness-of-fit tests in verification of simple and composite hypotheses is considered. Based on an investigation of the power, it is shown for the first time that there exist pairs of competing hypotheses which these tests are not able to distinguish in the case of small sample sizes n and type 1 error probabilities. It is shown that the reason for this lies in the bias of the tests in corresponding situations.