An Example of a Biased Two-Sided Ansari-Bradley Test for Dispersion with Equal Medians

Lehmann (1986) had a question whether the two-sided Wilcoxon test is unbiased. Sugiura (1965) gave a counterexample of a two-sided Wilcoxon test that is biased against some special two-sided translation alternatives. In this study, we extend the bias problem to the case of a linear rank statistics, especially for dispersion test. For a test function, we consider the most extreme case that two distributions have unequal dispersions. It can be visualized as an extreme example where the sample values, when ordered, fall in the pattern YYXXXXYY or XXYYYYXX. We will give a counterexample of a distribution-free Ansari- Bradley test for dispersion that is not unbiased, and show that the test is biased using Mathematica.